Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization

Abstract

We consider a wide class of summatory functions $F\left\{f;N,p^m\right\}={\sum }_{k\le N}f\left(p^{m}k\right)$, $m\in {\mathbb{Z}}_{+}\cup \left\{0\right\}$ associated with the multiplicative arithmetic functions f of a scaled variable $k\in {\mathbb{Z}}_{+}$, where p is a prime number. Assuming an asymptotic behavior of the summatory function, $F\{f;N,1\}\stackrel{N\to \infty }{=}{G}_1\left(N\right)\left[1+\mathcal{O}\left(G_2\left(N\right)\right)\right]$, where $G_1\left(N\right)=N^{a_1}{\left(\log N\right)}^{b_1}$, $G_2\left(N\right)=N^{-a_2}{\left(\log N\right)}^{-b_2}$ and $a_1,a_2\ge 0$, $-\infty <b_1,b_2<\infty$, we calculate the renormalization function $R\left(f;N,p^m\right)$, defined as a ratio $F\left\{f;N,p^m\right\}/F\{f;N,1\}$, and find its asymptotics $R_{\infty }\left(f;p^m\right)$ when $N\to \infty$. We prove that a renormalization function is multiplicative, i.e., $R_{\infty }\left(f;{\prod }_{i=1}^n{p}_i^{m_i}\right)={\prod }_{i=1}^n{R}_{\infty }\left(f;p_i^{m_i}\right)$ with n distinct primes $p_i$. We extend these results to the other summatory functions ${\sum }_{k\le N}f\left(p^m{k}^l\right)$, $m,l,k\in {\mathbb{Z}}_{+}$ and ${\sum }_{k\le N}{\prod }_{i=1}^n{f}_i\left(k{p}^{m_i}\right)$, $f_i\ne f_j$, $m_i\ne m_j$. We apply the derived formulas to a large number of basic summatory functions including the Euler $\varphi \left(k\right)$ and Dedekind $\psi \left(k\right)$ totient functions, divisor ${\sigma }_n\left(k\right)$ and prime divisor $\beta \left(k\right)$ functions, the Ramanujan sum $C_q\left(n\right)$ and Ramanujan $\tau$ Dirichlet series, and others.

1. Summatory Multiplicative Functions with Scaled Variable

Among summatory arithmetic functions ${\sum }_{k\le N}f\left(k\right),k\in {\mathbb{Z}}_{+}$ of various $f\left(k\right)$, the most studied are the basic multiplicative functions $f\left(k\right)$ and their algebraic combinations. A study of different summatory functions and their asymptotics has a long history [1,2,3]. Their list includes totient functions: the Euler $\varphi \left(k\right)$, Dedekind $\psi \left(k\right)$ and Jordan $J_n\left(k\right)$, and nontotient functions: the Möbius $\mu \left(k\right)$ and the n-order Möbius ${\mu }_n\left(k\right)$, Liouville $\lambda \left(k\right)$, Piltz $d_n\left(k\right)$, divisor ${\sigma }_n\left(k\right)$, prime divisor $\beta \left(k\right)$, nonisomorphic Abelian group enumeration function $\alpha \left(k\right)$, exponentiation of additive functions $\omega \left(k\right)$, and $\mathrm{\Omega }\left(k\right)$, which give the numbers of distinct prime dividing k and total prime factors of k counted with multiplicities, respectively. The whole family of multiplicative arithmetic functions is much wider, e.g., the number $q_n\left(k\right)$ of representations of k by sum of two integral nth powers [4], the number $r_n\left(k\right)$ of representations of k by sum of n integer squares [3], the Legendre and Zsigmondy totient functions [2] and the Nagell totient function [5], the nonisomorphic solvable [6] and nilpotent [7] finite group enumeration functions, the Gauss [3], Ramanujan [3] and Kloosterman [8] sums, the Ramanujan $\tau \left(k\right)$ function [9], and other arithmetic functions studied in the last decade [10,11,12,13,14,15].

In this paper, we study a family of summatory multiplicative arithmetic functions with a scaled summation variable,

$$F\left\{f;N,p^m\right\}=\sum _{k\le N}f\left(p^{m}k\right), k,m\in {\mathbb{Z}}_{+}, p\mathrm{is} \mathrm{a} \mathrm{prime} \mathrm{number},$$

(1)

that, to the best of our knowledge, has not been discussed in the literature. For this reason, henceforth, we use the notation $F\{f;N,1\}$ for an unscaled summatory function. A description of asymptotics of $F\{f;N,1\}$, $N\to \infty$, assumes that we know two characteristics, its leading and error terms, $G_1\left(N\right)$ and $G_2\left(N\right)$, respectively, i.e.,

$$F\{f;N,1\}\stackrel{N\to \infty }{\eqsim }{G}_1\left(N\right)\left[1+\mathcal{O}\left(G_2\left(N\right)\right)\right].$$

(2)

A study of summatory functions $F\left\{f;N,p^m\right\}$ was motivated by an idea that came from the modern theory of critical phenomena (Kadanoff [16], Wilson [17]) and dealt with statistical sums for systems, characterized by different length scales. Instead of straightforward calculations, an approach was proposed that links the sums of two adjacent scales by a renormalization difference equation. In this paper, for the first time, we apply the ideas of renormalization from statistical physics to calculate asymptotically summatory multiplicative arithmetic functions.

This article is organized into six sections and concludes with concluding remarks. In Section 1.1, we introduce universality classes $\mathbb{B}\{G_1\left(N\right);G_2\left(N\right)\}$ of arithmetic functions $f\left(k\right)$ such that different $f\left(k\right)$ have the same $G_1\left(N\right)$ and $G_2\left(N\right)$. Inspecting a vast number of multiplicative functions $f\left(k\right)$, we focus on wide classes, $G_1\left(N\right)=N^{a_1}{\left(\log N\right)}^{b_1}$, $G_2\left(N\right)=N^{-a_2}{\left(\log N\right)}^{-b_2}$, where $a_1,a_2\ge 0$ and $-\infty <b_1,b_2<\infty$. In Section 1.2, we derive a functional equation defined at different scales $p^r$,

$$\begin{array}{c}\hfill F\left\{f;N,p^m\right\}=\sum _{r=0}^{⌊{\log }_{p}N⌋}{L}_r\left(f;p^m\right)F\left\{f;⌊{\displaystyle \frac{N}{p^r}}⌋,1\right\},\end{array}$$

(3)

where the characteristic functions $L_r\left(f;p^m\right)$ are satisfied by the recursive equations,

$$\begin{array}{c}\hfill L_r\left(f;p^m\right)=f\left(p^{m+r}\right)-\sum _{j=0}^{r-1}{L}_j\left(f;p^m\right)f\left(p^{r-j}\right), L_0\left(f;p^m\right)=f\left(p^m\right), f\left(1\right)=1,\end{array}$$

(4)

and $⌊u⌋$ denotes the largest integer not exceeding u. The functions $L_r\left(f;p^m\right)$ are calculated in (17) and their behavior in r is crucial for the convergence of numerical series. This is a subject of special discussion in the next section.

In Section 2, we define the renormalization function $R\left(f;N,p^m\right)$ and its asymptotics,

$$\begin{array}{c}\hfill R\left(f;N,p^m\right)={\displaystyle \frac{F\left\{f;N,p^m\right\}}{F\left\{f;N,1\right\}}},\hspace{1em}\hspace{1em}{R}_{\infty }\left(f,p^m\right)=\underset{N\to \infty }{\lim }R\left(f;N,p^m\right).\end{array}$$

(5)

The aim of this paper is to study the asymptotic renormalization function $R_{\infty }\left(f,p^m\right)$ in various aspects: (a) its existence as a convergent numerical series, (b) its multiplicativity property without specifying the function $f\left(k\right)$, (c) formulas for $R_{\infty }\left(f\left(k\right)·k^{-s},p^m\right)$ for corresponding Dirichlet series, (d) formulas for $R_{\infty }\left(f\left(k^n\right),p^m\right)$ for different arithmetic functions $f\left(k\right)$, (e) formulas $R_{\infty }\left(f,p^m\right)$ for basic arithmetic function $f\left(k\right)$. In short, we often omit the word ‘asymptotic’ and refer to $R_{\infty }\left(f,p^m\right)$ as a renormalization function unless this would mislead the readers.

By imposing restrictions on $L_r\left(f;p^m\right)$, in Section 2.1, we prove two lemmas on the convergence of numerical series and calculate the asymptotics of renormalization functions. In Section 2.2, by these constraints, we show that the error term $G_2\left(N\right)$ does not contribute to $R_{\infty }\left(f,p^m\right)$. In Section 2.3, we give a rational representation for $R_{\infty }\left(f,p^m\right)$, which is much easier to implement in analytic calculations.

In Section 3, we prove that the renormalization function has a multiplicative property in the following sense, $R_{\infty }\left(f;{\prod }_{i=1}^n{p}_i^{m_i}\right)={\prod }_{i=1}^n{R}_{\infty }\left(f;p_i^{m_i}\right)$ with n distinct primes $p_i\ge 2$ (see Corollary 1). Using the renormalization approach, we also calculate the summatory functions ${\sum }_{k_1,k_2\le N}f\left(k_1{k}_2\right)$.

In Section 4, we extend the renormalization approach on summatory ${\sum }_{k\le N}{\prod }_{i=1}^n{f}_i\left(k{p}^{m_i}\right)$, where $f_i\ne f_j$, $m_i\ne m_j$, and the corresponding Dirichlet series ${\sum }_{k=1}^{\infty }f\left(k{p}^m\right)k^{-s}$, $k,n\in {\mathbb{Z}}_{+}$. We also study the renormalization of the summatory function ${\sum }_{k\le N}f\left(k^n{p}^m\right)$.

In Section 5, we apply formulas, derived in Section 2, to calculate $R_{\infty }\{f;p^m\}$ for basic multiplicative arithmetic functions $f\left(k\right)$ and their combinations. Almost all summatory functions are considered by Theorem 2 based on a simple calculation of $f\left(p^r\right)$ and avoiding the cumbersome calculation of characteristic functions $L_r\left(f;p^m\right)$. The renormalization functions are given by algebraic and non-algebraic expressions as well, e.g., see $R_{\infty }\left({\sigma }_0^n;p^m\right)$ in (131) and $R_{\infty }\left({\sigma }_1/{\sigma }_0;p\right)$ in (135), respectively. In this regard, the Ramanujan $\tau$– function is of particular interest: unlike many other functions $f\left(k\right)$, its value for $k=p^r$ is given by a complex formula (151), while the characteristic functions $L_r\left(\tau ;p^m\right)$ and $L_r\left({\tau }^2;p^m\right)$ have been calculated in a simple form suitable for the explicit calculation of $R_{\infty }\left(\tau ·k^{-s},p^m\right)$ and $R_{\infty }\left({\tau }^2,p^m\right)$. We have found a new identity (157) for the Ramanujan $\tau$– function.

In Section 6, we give a numerical verification of the renormalization approach developed in the article using numerical calculations and show its validity with high precision.

1.1. Asymptotic Growth of Summatory Functions

Consider the summatory multiplicative arithmetic function $F\{f;N,1\}$ and represent its asymptotics in N, instead of (2), by using one constant $\mathcal{F}$ and two positive definite functions $G_1\left(N\right)$ and $G_2\left(N\right)$,

$$\begin{array}{ccc}& & F\{f;N,1\}\stackrel{N\to \infty }{=}\mathcal{F}{G}_1\left(N\right)\left[1+\mathcal{O}\left(G_2\left(N\right)\right)\right],\hfill \\ & & \underset{N\to \infty }{\lim }{G}_2\left(N\right)=0,\hspace{1em}{G}_1\left(N\right),G_2\left(N\right)>0,\hfill \end{array}$$

(6)

where “$\mathcal{O}$” stands for the “big–O” Landau symbol.

The asymptotic growth of $F\{f;N,1\}$ is determined by its leading term and is given by the nondecreasing function $G_1\left(N\right)$, that is, either increasing or equal to one, while the decreasing function $G_2\left(N\right)$ denotes the error term. The constant $\mathcal{F}$ is introduced to distinguish summatories with similar functions $G_1\left(N\right)$ and $G_2\left(N\right)$, e.g., see $F\{1/\varphi ;N,1\}$, $F\{1/\psi ;N,1\}$ and $F\{1/{\sigma }_1;N,1\}$ in Table 1.

Table 1. Summatory multiplicative arithmetic functions and their asymptotics.

$\mathit{f}\left(\mathit{k}\right)$	$\mathcal{F}$	${\mathit{G}}_1\left(\mathit{N}\right)$	${\mathit{G}}_2\left(\mathit{N}\right)$	Ref
${\mu }^2\left(k\right)$	${\zeta }^{-1}\left(2\right)$	N	$N^{-1/2}$	[18]
${\mu }^2\left(k\right)/\varphi \left(k\right)$	1	$\log N$	${\log }^{-1}N$	[19]
${\mu }_n\left(k\right)$	$A_n$, $n\ge 2$	N	$N^{{\displaystyle \frac{1}{n}}-1}\log N$	[20], p.193
$1/\varphi \left(k\right)$	$\zeta \left(2\right)\zeta \left(3\right)/\zeta \left(6\right)$	$\log N$	${\log }^{-1}N$	[21]
$k/\varphi \left(k\right)$	$\zeta \left(2\right)\zeta \left(3\right)/\zeta \left(6\right)$	N	$N^{-1}\log N$	[22]
$\psi \left(k\right)$	$1/2\zeta \left(2\right)/\zeta \left(4\right)$	$N^2$	$N^{-1}\log N$	[1], p.72
$1/\psi \left(k\right)$	$C_1\simeq 0.37396$	$\log N$	${\log }^{-1}N$	[23]
${\sigma }_{-b}\left(k\right)$	$\zeta (b+1)$	N, $b>0$, $b\ne 1$	$N^{-min\{1,b\}}$	[1], p.61
${\sigma }_{-1}\left(k\right)$	$\zeta \left(2\right)$	N	$N^{-1}\log N$	[1], p.61
${\sigma }_a\left(k\right)$	$\zeta (a+1)/(a+1)$	$N^{a+1}$, $a>0$, $a\ne 1$	$N^{-min\{1,a\}}$	[1], p.60
${\sigma }_1\left(k\right)$	$\zeta \left(2\right)/2$	$N^2$	$N^{-1}{\log }^{2/3}N$	[24]
${\sigma }_a^2\left(k\right)$	${\displaystyle \frac{{\zeta }^2(1+|a\left|\right)\zeta (1+2|a\left|\right)}{\zeta (2+2|a\left|\right)}}$	$N^{1+a+\left|a\right|}$, $0<\left|a\right|<1$	$N^{-\left|a\right|}\log N$	[25]
${\sigma }_1^2\left(k\right)$	$5/6\zeta \left(3\right)$	$N^3$	$N^{-1}{\log }^{5/3}N$	[26]
${\sigma }_0^n\left(k\right)$	$D_n$, $D_1=1$, $D_2={\pi }^{-2}$	$N·{\log }^{2^n-1}N$	${\log }^{-1}N$	[27,28]
$1/{\sigma }_0\left(k\right)$	$D_{-1}\simeq 0.5469$	$N·{\log }^{-1/2}N$	${\log }^{-1}N$	[27,28]
${\sigma }_1\left(k\right)/\varphi \left(k\right)$	$C_2\simeq 3.6174 \star$	N	$N^{-1}{\log }^{2}N$	[25]
${\sigma }_1\left(k\right)/{\sigma }_0\left(k\right)$	$C_3\simeq 0.3569$	$N^2{\log }^{-1/2}N$	${\log }^{-1}N$	[29]
$1/{\sigma }_1\left(k\right)$	$C_4\simeq 0.6728$	$\log N$	${\log }^{-1}N$	[23]
$d_n\left(k\right)$	$1/\Gamma \left(n\right)$	$N·{\log }^{n-1}N$	${\log }^{-1}N$	[30,31]
$d_n^2\left(k\right)$	$E_n$, $E_2=D_2$	$N·{\log }^{n^2-1}N$	${\log }^{-1}N$	[32]
$1/d_n\left(k\right)$	$K_n$, $K_2=D_{-1}$	$N·{\log }^{1/n-1}N$	${\log }^{-1}N$	[33]
$\beta \left(k\right)$	$\zeta \left(2\right)\zeta \left(3\right)/\zeta \left(6\right)$	N	$N^{-1/2}$	[34,35]
$\alpha \left(k\right)$	${\prod }_{l=2}^{\infty }\zeta \left(l\right)\simeq 2.29486$	N	$N^{-1/2}$	[36]
$1/\alpha \left(k\right)$	$C_5\simeq 0.75204$	N	$N^{-1/2}{\log }^{-1/2}N$	[37], p.16
$2^{\omega \left(k\right)}$	${\zeta }^{-1}\left(2\right)$	$N·\log N$	${\log }^{-1}N$	[31]
$3^{\omega \left(k\right)}$	$C_6\simeq 0.14338$	$N·{\log }^{2}N$	${\log }^{-1}N$	[38], p.53
$2^{\mathrm{\Omega }\left(k\right)}$	$C_7\simeq 0.27317$	$N·{\log }^{2}N$	${\log }^{-1}N$	[39]
$q_n\left(k\right)$	$2{\Gamma }^2\left(n^{-1}\right)/\left(n\Gamma \left(2n^{-1}\right)\right)$	$N^{2/n}$, $n\ge 3$	$N^{-1/n(1+1/n)}$	[4], p.143
$r_2\left(k\right)=q_2\left(k\right)$	$\pi$	N	$N^{-1/2}$	[3]
$\left|\tau \right(k\left)\right|$	$C_8\simeq 0.0996 \star$	$N^{13/2}{\log }^{-1+8/\left(3\pi \right)}N$	?	[40,41]
${\tau }^2\left(k\right)$	$C_9\simeq 0.032007$	$N^{12}$	?	[9]
${\tau }^4\left(k\right)$	$C_{10}\simeq 0.0026 \star$	$N^{23}\log N$	?	[41]

Different arithmetic functions $f\left(k\right)$ can have the same $G_1\left(N\right)$ and $G_2\left(N\right)$; therefore, the entire set of $f\left(k\right)$ can be decomposed into different universality classes $\mathbb{B}\{G_1\left(N\right);G_2\left(N\right)\}$ as follows:

$$\begin{array}{c}\hfill f\left(k\right)\in \mathbb{B}\{G_1\left(N\right);G_2\left(N\right)\},\end{array}$$

(7)

where

$$\begin{array}{ccc}& & \mathbb{B}\{G_1\left(N\right);G_2\left(N\right)\}=\hfill \\ & & \left\{f\left(k\right)\left|{\displaystyle \frac{F\{f;N,1\}}{\mathcal{F}{G}_1\left(N\right)}}\right.\stackrel{N\to \infty }{=}1,\left|{\displaystyle \frac{F\{f;N,1\}}{\mathcal{F}{G}_1\left(N\right)}}-1\right|\stackrel{N\to \infty }{<}\mathcal{C}{G}_2\left(N\right);0<\mathcal{C},\mathcal{F}<\infty \right\}.\hfill \end{array}$$

Below are examples of various multiplicative functions $f\left(k\right)$ belonging to different universality classes.

$$\begin{array}{lll}{\displaystyle \frac{1}{\varphi \left(k\right)}},\hspace{1em}\hfill & {\displaystyle \frac{1}{\psi \left(k\right)}},\hspace{1em}& {\displaystyle \frac{\varphi \left(k\right)}{k^2}},\hspace{1em}{\displaystyle \frac{1}{{\sigma }_1\left(k\right)}},\hspace{1em}{d}_1\left(k\right),\hspace{1em}{\displaystyle \frac{r_2\left(k\right)}{k}}\in \mathbb{B}\left\{\log N;{\displaystyle \frac{1}{\log N}}\right\},\\ {\displaystyle \frac{\varphi \left(k\right)}{k}},\hspace{1em}\hfill & {\displaystyle \frac{\psi \left(k\right)}{k}},\hspace{1em}& {\displaystyle \frac{{\sigma }_1\left(k\right)}{k}},\hspace{1em}{\sigma }_{-1}\left(k\right)\in \mathbb{B}\left\{N;{\displaystyle \frac{\log N}{N}}\right\},\\ {\sigma }_0\left(k\right),\hfill & 2^{\omega \left(k\right)},& d_2\left(k\right)\in \mathbb{B}\left\{N\log N;{\displaystyle \frac{1}{\log N}}\right\},\\ 3^{\omega \left(k\right)},\hfill & 2^{\mathrm{\Omega }\left(k\right)},& d_3\left(k\right)\in \mathbb{B}\left\{N{\log }^{2}N;{\displaystyle \frac{1}{\log N}}\right\},\\ {\mu }^2\left(k\right),\hfill & \alpha \left(k\right),\hspace{1em}& \beta \left(k\right),r_2\left(k\right)\in \mathbb{B}\left\{N;{\displaystyle \frac{1}{\sqrt{N}}}\right\}.\end{array}$$

By the inspection of a vast number of multiplicative functions $f\left(k\right)$ with known asymptotics $G_1\left(N\right)$ and $G_2\left(N\right)$ (see Table 1 and Table 2), in this paper, we focus on their most wide class,

$$\begin{array}{ccc}\hfill G_1\left(N\right)=N^{a_1}{\left(\log N\right)}^{b_1},& & \left\{\begin{array}{cc}{a}_1>0,\hfill & \hfill -\infty <b_1<\infty ,\\ a_1=0,\hfill & \hfill 0\le b_1<\infty ,\end{array}\right.\hfill \\ \hfill G_2\left(N\right)={\displaystyle \frac{1}{N^{a_2}{\left(\log N\right)}^{b_2}}},& & \left\{\begin{array}{cc}{a}_2>0,\hfill & \hfill -\infty <b_2<\infty ,\\ a_2=0,\hfill & \hfill 0<b_2<\infty .\end{array}\right.\hfill \end{array}$$

(8)

Table 2. Other summatory functions and the Dirichlet series of multiplicative functions.

$\mathit{f}\left(\mathit{k}\right)$	$\mathcal{F}$	${\mathit{G}}_1\left(\mathit{N}\right)$	${\mathit{G}}_2\left(\mathit{N}\right)$	Ref
$\mu \left(k\right)/k^s$	${\zeta }^{-1}\left(s\right)$, $s>1$	$N^0$	$N^{-(s-1)}$	[1], p.231
${\mu }^2\left(k\right)/k^s$	$\zeta \left(s\right)/\zeta \left(2s\right)$, $s>1$	$N^0$	$N^{-(s-1)}$	[1], p.241
${\mu }^2\left(k\right)/k$	${\zeta }^{-1}\left(2\right)$	$\log N$	${\log }^{-1}N$	[42], p.195
$\lambda \left(k\right)/k^s$	$\zeta \left(2s\right)/\zeta \left(s\right)$, $s>1$	$N^0$	$N^{-(s-1)}$	[1], p.231
$\varphi \left(k\right)/k^s$	${\zeta }^{-1}\left(2\right)/(2-s)$, $0<s\le 1$	$N^{2-s}$	$N^{-1}\log N$	[1], p.71
$\varphi \left(k\right)/k^s$	${\zeta }^{-1}\left(2\right)/(2-s)$, $1<s<2$	$N^{2-s}$	$N^{-(2-s)}$	[1], p.71
$\varphi \left(k\right)/k^2$	${\zeta }^{-1}\left(2\right)$	$\log N$	${\log }^{-1}N$	[1], p.71
$\varphi \left(k\right)/k^s$	$\zeta (s-1)/\zeta \left(s\right)$, $s>2$	$N^0$	$N^{-(s-2)}$	[1], p.231
${\left(\varphi \left(k\right)/k\right)}^s$	$B_s$, $B_1={\zeta }^{-1}\left(2\right)$	N, $s>0$	$N^{-1}{\log }^{s}N$	[43]
${\sigma }_0\left(k\right)/k^s$	${(1-s)}^{-1}$, $0<s<1$	$N^{1-s}\log N$	${\log }^{-1}N$	[1], p.70
${\sigma }_0\left(k\right)/k$	1/2	${\log }^{2}N$	${\log }^{-1}N$	[1], p.70
${\sigma }_0\left(k\right)/k^s$	${\zeta }^2\left(s\right)$, $s>1$	$N^0$	$N^{-(s-1)}\log N$	[1], p.231
${({\sigma }_1\left(k\right)/k)}^s$	$I_s$, $I_1=\zeta \left(2\right)$, $I_2={\displaystyle \frac{5}{2}}\zeta \left(3\right)$	N, $s>0$	$N^{-1}{\log }^{s}N$	[25,26]
$r_2\left(k\right)/k$	$\pi$	$\log N$	${\log }^{-1}N$	[3]
${\tau }^2\left(k\right)/k^{25/2}$	$C_{11}\simeq 1.58824$	$N^0$	?	[44]

Remark 1.

Table 1 and Table 2 do not present any example of multiplicative arithmetic functions $f\left(k\right)$ of special universality classes $\mathbb{B}\left\{N^{a_1}{\left(\log N\right)}^{b_1};N^{-a_2}{\left(\log N\right)}^{-b_2}\right\}$ such that

$$\begin{array}{ccc}& & 0<a_1<a_2, -\infty <b_1,b_2<\infty ,\hfill \\ & & a_1=0<a_2, 0\le b_1<b_2<\infty .\hfill \end{array}$$

(9)

Despite an extensive search in the available literature, we have not found such functions there.

In Table 1 and Table 2, we present a long list of multiplicative functions that belong to one of the universality classes $\mathbb{B}\left\{G_1\left(N\right);G_2\left(N\right)\right\}$ satisfying (8).

We use the standard notation for the functions mentioned in Section 1. Here, $\zeta \left(s\right)$ stands for the Riemann zeta function, the explicit expressions for $A_n$, $B_s$, $D_n$, $E_n$, $K_n$, $I_n$, and the values for $C_i$ are given in the corresponding references. The values of $C_2$, $C_8$, $C_{10}$ were calculated by the author and marked by (⋆). The error terms for summatories of $\left|\tau \right(k\left)\right|$, ${\tau }^2\left(k\right)$, ${\tau }^4\left(k\right)$ and ${\tau }^2\left(k\right)/\sqrt{k^{25}}$ are unknown to date.

The relation between the multiplicative properties of the arithmetic functions and the asymptotics of their summatory functions is not straightforward. In other words, the following correspondence:

$$\begin{array}{c}\hfill f\left(k\right)\hspace{1em}\mathrm{is}\mathrm{a}\mathrm{multiplicative}\mathrm{function}⟷f\left(k\right)\in \mathbb{B}\left\{N^{a_1}{\left(\log N\right)}^{b_1};N^{-a_2}{\left(\log N\right)}^{-b_2}\right\}\end{array}$$

(10)

is neither bijective nor injective. Indeed, the direction ‘⟵’ is not satisfied, since there exists a nonmultiplicative function $f\left(k\right)=\log \varphi \left(k\right)/\log {\sigma }_1\left(k\right)$, which has the summatory function $F\left\{f;N,1\right\}\stackrel{N\to \infty }{=}N+\mathcal{O}\left(N{\log }^{-1}N\right)$ [37].

Regarding another direction ‘⟶’, there exist multiplicative arithmetic functions with summatory growth that differ from $N^{a_1}{\left(\log N\right)}^{b_1}$ and come by enumerating finite groups. Let $\chi \left(k\right)$ be a number of nilpotent groups of order k, which is multiplicative, because each finite nilpotent group is a direct product of its Sylow subgroups [7]. When $k=p^r$, for a prime p, it is known [45] that $\chi \left(k\right)\simeq p^{(2/27+o\left(1\right))r^3}$, that is, of the order $k^{{\left({\log }_{p}k\right)}^2}$.

Consider its summatory function $F[\chi ;N,1]$, which is the number of nilpotent groups of the order of most N. For some $r>0$, we have $2^r\le N<2^{r+1}$. Then, for $0<s<1$ we get,

$$\begin{array}{ccc}& & F[\chi ;N,1]\ge \chi \left(2^r\right)=2^{(2/27+o\left(1\right))r^3}\ge {\left(2^{r+1}\right)}^{2s/27{(r+1)}^2} ⟶\hfill \\ & & ⟶ F[\chi ;N,1]\ge N^{2s/27{\left(\log N\right)}^2},\hfill \end{array}$$

(11)

and therefore, $\chi \left(k\right)\notin \mathbb{B}\left\{N^{a_1}{\left(\log N\right)}^{b_1};N^{-a_2}{\left(\log N\right)}^{-b_2}\right\}$. Note that the number $\chi (k,c,d)$, nilpotent groups of order k, nilpotency class at most n, generated by at most m elements, belongs to the universality class defined in (8), where $a_i$ and $b_i$ depend on n and m [46].

Other examples of multiplicative functions with summatory growth $N^{a_1}{\left(\log N\right)}^{b_1}$ and error terms, which differ from $N^{-a_2}{\left(\log N\right)}^{-b_2}$, are given in [24],

$$\begin{array}{c}\hfill \varphi \left(k\right)\in \mathbb{B}\left(N^2;{\displaystyle \frac{1}{N}}{\left(\log N\right)}^{2/3}{\left(\log \log N\right)}^{4/3}\right), {\displaystyle \frac{\varphi \left(k\right)}{k}}\in \mathbb{B}\left(N;{\displaystyle \frac{1}{N}}{\left(\log N\right)}^{2/3}{\left(\log \log N\right)}^{4/3}\right).\end{array}$$

They can also be encompassed within the universality classes by extending the latter on a much wider family of asymptotics, e.g.,

$$\mathbb{B}\left\{N^{a_1}{\left(\log N\right)}^{b_1}{\left(\log \log N\right)}^{c_1};N^{-a_2}{\left(\log N\right)}^{-b_2}{\left(\log \log N\right)}^{-c_2}\right\}.$$

Keeping in mind this option, we continue to study summatory multiplicative functions with universality classes of asymptotics, given in (8).

1.2. Scaling Equation for Summatory Functions

Represent $F\left\{f;N,p^m\right\}$ as a sum of two summatory functions

$$\begin{array}{c}\hfill F\left\{f;N,p^m\right\}=\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k=1}{p\nmid k}}}^{N}f\left(p^{m}k\right)+\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k=p}{p\mid k}}}^{N}f\left(p^{m}k\right).\end{array}$$

(12)

Making use of multiplicativity, $f\left(k_1{k}_2\right)=f\left(k_1\right)f\left(k_2\right)$ if $\gcd \left(k_1{k}_2\right)=1$, $f\left(1\right)=1$, we obtain

$$\begin{array}{ccc}\hfill F\left\{f;N,p^m\right\}& =& f\left(p^m\right)\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k=1}{p\nmid k}}}^{N}f\left(k\right)+\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k=p}{p\mid k}}}^{N}f\left(p^{m}k\right)=f\left(p^m\right)\left(\sum _{k=1}^{N}f\left(k\right)-\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k=p}{p\mid k}}}^{N}f\left(k\right)\right)+\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k=p}{p\mid k}}}^{N}f\left(p^{m}k\right)\hfill \\ & =& f\left(p^m\right)F\{f;N,1\}-f\left(p^m\right)\sum _{l=1}^{N_1}f\left(pl\right)+\sum _{l=1}^{N_1}f\left(p^{m+1}l\right)\hfill \\ & =& f\left(p^m\right)F\{f;N,1\}-f\left(p^m\right)F\{f;N_1,p\}+F\left\{f;N_1,p^{m+1}\right\}.\hfill \end{array}$$

(13)

where $N_r=⌊N/p^r⌋$. The recursion (13) holds for any $N_r$, i.e.,

$$\begin{array}{c}\hfill F\left\{f;N_r,p^m\right\}=f\left(p^m\right)F\{f;N_r,1\}-f\left(p^m\right)F\{f;N_{r+1},p\}+F\left\{f;N_{r+1},p^{m+1}\right\}.\end{array}$$

(14)

Substituting (14) into (13), we obtain

$$\begin{array}{ccc}& & F\left\{f;N,p^m\right\}=\hfill \\ & & f\left(p^m\right)F\{f;N,1\}-f\left(p^m\right)\left[f\left(p\right)F\{f;N_1,1\}-f\left(p\right)F\{f;N_2,p\}+F\left\{f;N_2,p^2\right\}\right]+\hfill \\ & & f\left(p^{m+1}\right)F\{f;N_1,1\}-f\left(p^{m+1}\right)F\{f;N_2,p\}+F\left\{f;N_2,p^{m+2}\right\}=\hfill \\ & & f\left(p^m\right)F\{f;N,1\}+\left[f\left(p^{m+1}\right)-f\left(p^m\right)f\left(p\right)\right]F\{f;N_1,1\}+\hfill \\ & & \left[f\left(p^m\right)f\left(p\right)-f\left(p^{m+1}\right)\right]F\{f;N_2,p\}-f\left(p^m\right)F\left\{f;N_2,p^2\right\}+F\left\{f;N_2,p^{m+2}\right\}.\hfill \end{array}$$

Continuing this procedure recursively, we finally obtain,

$$\begin{array}{ccc}& & F\left\{f;N,p^m\right\}=\sum _{r=0}^{⌊{\log }_{p}N⌋}{L}_r\left(f;p^m\right)F\{f;N_r,1\}, \mathrm{where}\hfill \end{array}$$

(15)

$$\begin{array}{ccc}& & L_r\left(f;p^m\right)=f\left(p^{m+r}\right)-\sum _{j=0}^{r-1}{L}_j\left(f;p^m\right)f\left(p^{r-j}\right), L_0\left(f;p^m\right)=f\left(p^m\right).\hfill \end{array}$$

(16)

The straightforward calculations of $L_r\left(f;p^m\right)$ give

$$\begin{array}{ccc}\hfill L_1\left(f;p^m\right)& =& f\left(p^{m+1}\right)-f\left(p^m\right)f\left(p\right),\hfill \\ \hfill L_2\left(f;p^m\right)& =& f\left(p^{m+2}\right)-f\left(p^m\right)f\left(p^2\right)-f\left(p^{m+1}\right)f\left(p\right)+f\left(p^m\right)f^2\left(p\right),\hfill \\ \hfill L_3\left(f;p^m\right)& =& f\left(p^{m+3}\right)-f\left(p^m\right)f\left(p^3\right)-f\left(p^{m+1}\right)f\left(p^2\right)+2f\left(p^m\right)f\left(p^2\right)f\left(p\right)-\hfill \\ & & f\left(p^{m+2}\right)f\left(p\right)+f\left(p^{m+1}\right)f^2\left(p\right)-f\left(p^m\right)f^3\left(p\right), \mathrm{etc},\hfill \end{array}$$

(17)

such that for $p=1$ or $m=0$, we obtain $L_0\left(f;1\right)=1$ and $L_r\left(f;1\right)=0$, $r\ge 1$.

By (16) and (17), the general formulas for $L_r\left(f;p^m\right)$ can be calculated by induction for simple arithmetic functions $f\left(k\right)$ and $1/f\left(k\right)$ such that $f\left(p^m\right)=A_f{p}^{m-1}$, $m\ge 1$,

$$\begin{array}{c}\hfill L_r\left(f;p^m\right)=A_f{(p-A_f)}^r{p}^{m-1}, L_r\left({\displaystyle \frac{1}{f}};p^m\right)={\displaystyle \frac{1}{A_f}}{\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{A_f}}\right)}^r{\displaystyle \frac{1}{p^{m-1}}}, r\ge 1, \end{array}$$

(18)

and $A_f$ denotes a real constant, e.g., $A_{\varphi }=p-1$, $A_{\psi }=p+1$ that gives

$$\begin{array}{ccc}& & L_r\left(\varphi ;p^m\right)=p^{m-1}(p-1),\hspace{1em}{L}_r\left({\displaystyle \frac{1}{\varphi }};p^m\right)={\displaystyle \frac{{(-1)}^r{p}^{1-r-m}}{{(p-1)}^{r+1}}},\hfill \\ & & L_r\left(\psi ;p^m\right)={(-1)}^r{p}^{m-1}(p+1), L_r\left({\displaystyle \frac{1}{\psi }};p^m\right)={\displaystyle \frac{p^{1-r-m}}{{(p+1)}^{r+1}}}.\hfill \end{array}$$

Another example of arithmetic functions that result in $L_r\left(f;p^m\right)=0$, $r\ge 1$, are those where $A_f=p$ in (18) or $f\left(p^m\right)=c^m$, e.g., $2^{\mathrm{\Omega }\left(p^m\right)}=2^m$ and $\lambda \left(p^m\right)={(-1)}^m$.

In general case of $f\left(k\right)$, the formula for $L_r\left(f;p^m\right)$ for an arbitrary $r\ge 0$ may be difficult to recognize from its partial expressions, e.g., for $f\left(k\right)=1/{\sigma }_0\left(k\right)$,

$$\begin{array}{ccc}& & L_0={\displaystyle \frac{1}{m+1}},\hspace{1em}{L}_1={\displaystyle \frac{mm!}{2(m+2)!}},\hspace{1em}{L}_2={\displaystyle \frac{(5m+7)mm!}{12(m+3)!}},\hfill \\ & & L_3={\displaystyle \frac{(9m^2+35m+32)mm!}{24(m+4)!}}.\hfill \end{array}$$

(19)

In Section 5.2.1, Formula (133), we show that $L_r$ in (19) come as coefficients in the series expansion of the function, involving logarithmic and hypergeometric functions.

Remark 2.

Consider an integer N in the range $p^{\overline{r}}\le N<p^{\overline{r}+1}$ where $\overline{r}=⌊{\log }_{p}N⌋$ and write (15) as follows,

$$\begin{array}{c}\hfill F\left\{f;N,p^m\right\}=\sum _{r=0}^{\overline{r}-1}{L}_r\left(f;p^m\right)F\{f;N_r,1\}+L_{\overline{r}}\left(f;p^m\right)F\{f;N_{\overline{r}},1\},\end{array}$$

(20)

where $1\le N_{\overline{r}}=⌊N/p^{\overline{r}}⌋<p$ and $F[f;N_{\overline{r}},1]={\sum }_{k=1}^{N_{\overline{r}}}f\left(k\right)<{\sum }_{k=1}^{p}f\left(k\right)$ is a finite number.

We make use of the representation (20) in Section 2.1 when studying the asymptotics of renormalization functions for universality classes with $b_1<0$ (Lemma 2) and $b_1<b_2$ (Lemma 4).

2. Renormalization Function with Simple Scaling

Define the renormalization function as a ratio of two summatory functions

$$\begin{array}{c}\hfill R\left(f;N,p^m\right)={\displaystyle \frac{F\left\{f;N,p^m\right\}}{F\{f;N,1\}}},\hspace{1em}R\left(f;N,1\right)=1.\end{array}$$

(21)

Substituting (6) into (21), we obtain its asymptotic behavior

$$\begin{array}{c}\hfill R\left(f;N,p^m\right)\stackrel{N\to \infty }{=}{\displaystyle \frac{{\mathcal{R}}_1\left(f;N,p^m\right)}{1+\mathcal{O}\left(G_2\left(N\right)\right)}}+{\displaystyle \frac{{\mathcal{R}}_2\left(f;N,p^m\right)}{1+\mathcal{O}\left(G_2\left(N\right)\right)}},\end{array}$$

where ${\mathcal{R}}_1\left(f;N,p^m\right)$ and ${\mathcal{R}}_2\left(f;N,p^m\right)$ are defined according to (15) as follows:

$$\begin{array}{ccc}\hfill {\mathcal{R}}_1\left(f;N,p^m\right)& =& \sum _{r=0}^{⌊{\log }_{p}N⌋}{L}_r\left(f;p^m\right){\displaystyle \frac{G_1\left(N_r\right)}{G_1\left(N\right)}},\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill {\mathcal{R}}_2\left(f;N,p^m\right)& =& \sum _{r=0}^{⌊{\log }_{p}N⌋}{L}_r\left(f;p^m\right){\displaystyle \frac{G_1\left(N_r\right)}{G_1\left(N\right)}}\mathcal{O}\left(G_2\left(N_r\right)\right).\hfill \end{array}$$

(23)

If both numerical series ${\mathcal{R}}_1\left(f;N,p^m\right)$ and ${\mathcal{R}}_2\left(f;N,p^m\right)$ converge when $N\to \infty$, then

$$\begin{array}{c}\hfill \underset{N\to \infty }{\lim }R\left(f;N,p^m\right):=R_{\infty }(f;p^m)=\underset{N\to \infty }{\lim }{\mathcal{R}}_1\left(f;N,p^m\right)+\underset{N\to \infty }{\lim }{\mathcal{R}}_2\left(f;N,p^m\right).\end{array}$$

(24)

What can be said about the convergence of $R\left(f;N,p^m\right)$ without knowing exactly the multiplicative function $f\left(k\right)$? Formulas (22) and (23) for ${\mathcal{R}}_1\left(f;N,p^m\right)$ and ${\mathcal{R}}_2\left(f;N,p^m\right)$ indicate that most of the information is hidden in the asymptotics $G_1\left(N\right)$ and $G_2\left(N\right)$.

Substitute $G_1\left(N\right)=N^{a_1}{\left(\log N\right)}^{b_1}$ into (22) and obtain

$$\begin{array}{c}\hfill {\mathcal{R}}_1\left(f;N,p^m\right)=\sum _{r=0}^{⌊{\log }_{p}N⌋}{\displaystyle \frac{L_r\left(f;p^m\right)}{p^{a_{1}r}}}{\left(1-{\displaystyle \frac{r}{{\log }_{p}N}}\right)}^{b_1}.\end{array}$$

(25)

Regarding ${\mathcal{R}}_2\left(f;N,p^m\right)$, which is responsible for the error term in $R\left(f;N,p^m\right)$, note that according to the definition (7), we obtain $\mathcal{O}\left(G_2\left(N\right)\right)\le \mathcal{C}{G}_2\left(N\right)$. Applying this inequality to Formula (23),

$$\begin{array}{c}\hfill \left|{\mathcal{R}}_2\left(f;N,p^m\right)\right|\le \mathcal{C}\sum _{r=0}^{⌊{\log }_{p}N⌋}\left|L_r\left(f;p^m\right)\right|{\displaystyle \frac{G_1\left(N_r\right)}{G_1\left(N\right)}}{G}_2\left(N_r\right),\end{array}$$

and substituting $G_2\left(N\right)=N^{-a_2}{\left(\log N\right)}^{-b_2}$ into the last expression, we obtain an estimate,

$$\begin{array}{c}\hfill \left|{\mathcal{R}}_2\left(f;N,p^m\right)\right|\le {\displaystyle \frac{\mathcal{C}}{N^{a_2}{\left(\log N\right)}^{b_2}}}{\mathcal{R}}_3\left(f;N,p^m\right), \mathrm{where}\end{array}$$

(26)

$$\begin{array}{c}\hfill {\mathcal{R}}_3\left(f;N,p^m\right)=\sum _{r=0}^{⌊{\log }_{p}N⌋}{\displaystyle \frac{\left|L_r\left(f;p^m\right)\right|}{p^{r(a_1-a_2)}}}{\left(1-{\displaystyle \frac{r}{{\log }_{p}N}}\right)}^{b_1-b_2}.\end{array}$$

(27)

Section 2.1 and Section 2.2 present a detailed analysis of the convergence of ${\mathcal{R}}_1\left(f;N,p^m\right)$ and ${\mathcal{R}}_2\left(f;N,p^m\right)$ for multiplicative functions $f\left(k\right)$ of several universality classes. We start with a certain class of $f\left(k\right)$ and, assuming only $L_r\left(f;p^m\right)\ge 0$, prove that the convergence of ${\mathcal{R}}_1\left(f;N,p^m\right)$ implies the convergence of ${\mathcal{R}}_2\left(f;N,p^m\right)$ to zero.

Proposition 1.

Let $f\left(k\right)\in \mathbb{B}\left\{N^{a_1}{\left(\log N\right)}^{b_1};N^{-a_1}{\left(\log N\right)}^{-b_2}\right\}$ be given in such a way that $L_r\left(f;p^m\right)\ge 0$, and let ${\mathcal{R}}_1\left(f;N,p^m\right)$ be convergent. Then, ${\mathcal{R}}_2\left(f;N,p^m\right)\stackrel{N\to \infty }{⟶}0$, in each of the cases,

$$\begin{array}{ccc}& & 1) a_1=a_2=0, b_1\ge b_2>0,\hfill \\ & & 2) 0<a_2<a_1, 0\le b_2\le b_1 or 0=a_2<a_1, 0<b_2\le b_1,\hfill \\ & & 3) 0<a_2<a_1, b_1<b_2, b_1<0 or 0=a_2<a_1, b_1<0<b_2.\hfill \end{array}$$

Proof. 

Keeping in mind $L_r\left(f;p^m\right)\ge 0$ and comparing (25) and (27), we conclude that if ${\mathcal{R}}_1\left(f;N,p^m\right)$ is convergent when $a_1=0$ and $b_1\ge 0$, then ${\mathcal{R}}_3\left(f;N,p^m\right)$ is also convergent when $a_1=a_2=0$ and $b_1\ge b_2$. Substituting this into (26), we obtain

$$\begin{array}{c}\hfill \left|{\mathcal{R}}_2\left(f;N,p^m\right)\right|\le \mathcal{C}{\left(\log N\right)}^{-b_2}{\mathcal{R}}_3\left(f;N,p^m\right),\end{array}$$

(28)

which completes the proof if $b_2>0$. Indeed, if $b_2$ is positive and ${\mathcal{R}}_3\left(f;N,p^m\right)$ converges, then the r.h.s. in (28) converges to zero as does ${\mathcal{R}}_2\left(f;N,p^m\right)$. Applying similar reasoning to the other two cases (2) and (3), we completely prove the proposition. □

Table 1 shows when the first item in Proposition 1 can be applied: this is the inverse Dedekind function: $a_1=a_2=0$, $b_1=b_2=1$ and by (18) $L_r\left(1/\psi ;p^m\right)\ge 0$. However, neither the Euler totient function nor its inverse can be studied using Proposition 1, which is a rather weak statement.

The convergence of ${\mathcal{R}}_1\left(f;N,p^m\right)$ implies a zero limit of ${\mathcal{R}}_2\left(f;N,p^m\right)$, $N\to \infty$, over a much wider range of varying degrees $a_1,b_1$ and $a_2,b_2$. Indeed, to ensure that ${\mathcal{R}}_2\left(f;N,p^m\right)$ converges to zero, it is not necessary to require ${\mathcal{R}}_3\left(f;N,p^m\right)$ to converge in (26), but it is sufficient to allow ${\mathcal{R}}_3\left(f;N,p^m\right)$ to grow at a rate less than $N^{a_2}{\left(\log N\right)}^{b_2}$. However, this would require more assumptions about $L_r\left(f;p^m\right)$. In the following Section 2.1 and Section 2.2, we study the convergence problem in more detail and prove the main result of this section in Theorem 1.

2.1. Convergence of ${\mathcal{R}}_1\left(f;N,p^m\right)$

In this section, we study the convergence of ${\mathcal{R}}_1\left(f;N,p^m\right)$ when $N\to \infty$. The conditions imposed on $L_r\left(f;p^m\right)$ provide convergence of ${\mathcal{R}}_1\left(f;N,p^m\right)$. In this section and the next, we repeatedly make use of the squeeze ($\mathsf{S}\mathsf{Q}$) theorem [47], which is also known as the pinching or sandwich theorems.

Lemma 1.

Let a function $f\left(k\right)\in \mathbb{B}\left\{N^{a_1}{\left(\log N\right)}^{b_1};G_2\left(N\right)\right\}$, $a_1\ge 0$, $b_1\ge 0$, be given and let there exist two numbers $\mathcal{K}>0$ and $\gamma <a_1$ and an integer $r_{*}\ge 0$ such that $|L_r\left(f;p^m\right)|\le \mathcal{K}{p}^{\gamma r}$ for all $r\ge r_{*}$. Then,

$$\begin{array}{c}\hfill \underset{N\to \infty }{\lim }{\mathcal{R}}_1\left(f;N,p^m\right)=\sum _{r=0}^{\infty }{\displaystyle \frac{L_r\left(f;p^m\right)}{p^{a_{1}r}}}.\end{array}$$

(29)

Proof. 

First, consider ${\mathcal{R}}_1\left(f;N,p^m\right)$ given in (25) when $a_1\ge 0$, $b_1\in {\mathbb{Z}}_{+}\cup \left\{0\right\}$, i.e., $b_1$ is a nonnegative integer. After binomial expansion in (25), we obtain

$$\begin{array}{c}\hfill {\mathcal{R}}_1\left(f;N,p^m\right)=\sum _{r=0}^{⌊{\log }_{p}N⌋}{\displaystyle \frac{L_r\left(f;p^m\right)}{p^{a_{1}r}}}+\sum _{k=1}^{b_1}{\left({\displaystyle \frac{-1}{{\log }_{p}N}}\right)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{b_1}{k}}\right)\sum _{r=0}^{⌊{\log }_{p}N⌋}{\displaystyle \frac{L_r\left(f;p^m\right)}{p^{a_{1}r}}}{r}^k.\end{array}$$

(30)

Focus on the inner sum

$$\begin{array}{c}\hfill {\mathcal{R}}_4\left(f;N,p^m,k\right)={\displaystyle \frac{1}{{\left({\log }_{p}N\right)}^k}}\sum _{r=0}^{⌊{\log }_{p}N⌋}{\displaystyle \frac{L_r\left(f;p^m\right)}{p^{a_{1}r}}}{r}^k, 1\le k\le b_1,\end{array}$$

(31)

and prove that the sum in (31) is convergent absolutely.

To find an estimate for ${\mathcal{R}}_4\left(f;N,p^m,k\right)$, we must consider the last sum in the interval $\left(r_{*},⌊{\log }_{p}N⌋\right)$ where inequality $|L_r\left(f;p^m\right)|\le \mathcal{K}{p}^{\gamma r}$ holds. However, due to a prefactor ${\left({\log }_{p}N\right)}^{-k}$, $k\ge 1$, summation over the interval $(0,r_{*}-1)$ does not contribute to the limit as $N\to \infty$ and does not change the convergence of the entire sum (31). Then,

$$\begin{array}{c}\hfill |{\mathcal{R}}_4\left(f;N,p^m,k\right)|\le {\displaystyle \frac{1}{{\left({\log }_{p}N\right)}^k}}\sum _{r=0}^{⌊{\log }_{p}N⌋}{\displaystyle \frac{|L_r\left(f;p^m\right)|}{p^{a_{1}r}}}{r}^k\le {\displaystyle \frac{\mathcal{K}}{{\left({\log }_{p}N\right)}^k}}\sum _{r=0}^{⌊{\log }_{p}N⌋}{\displaystyle \frac{r^k}{p^{ϵr}}},\end{array}$$

(32)

where $ϵ=a_1-\gamma >0$. Denote $M=⌊{\log }_{p}N⌋$ and consider the sum in the r.h.s. of (32),

$$\begin{array}{c}\hfill T(p,k,ϵ,M)=\sum _{r=0}^M{\displaystyle \frac{r^k}{p^{ϵr}}}={\mathrm{Li}}_{-k}\left(p^{-ϵ}\right)-p^{-ϵ(M+1)}\mathrm{\Phi }\left(p^{-ϵ},-k,M+1\right),\end{array}$$

(33)

where ${\mathrm{Li}}_s\left(z\right)$ and $\mathrm{\Phi }(z,s,a)$ are the polylogarithm and the Hurwitz–Lerch zeta function [48], respectively,

$$\begin{array}{c}\hfill {\mathrm{Li}}_s\left(z\right)=\sum _{k=1}^{\infty }{\displaystyle \frac{z^k}{k^s}},\mathrm{\Phi }(z,s,a)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{{(a+k)}^s}}.\end{array}$$

(34)

Keeping in mind the asymptotics of $\mathrm{\Phi }(z,s,a)$, $a\to \infty$, for fixed s and z (Thm.1, [49]), $\mathrm{\Phi }(z,s,a)\simeq a^{-s}/(1-z)$, and combining it with (31) and (33), we get

$$\begin{array}{c}\hfill |{\mathcal{R}}_4\left(f;N,p^m,k\right)|\stackrel{N\to \infty }{\le }\mathcal{K}\left({\displaystyle \frac{{\mathrm{Li}}_{-k}\left(p^{-ϵ}\right)}{{\left({\log }_{p}N\right)}^k}}-{\displaystyle \frac{N^{-ϵ}}{p^{ϵ}-1}}\right), 1\le k\le b_1.\end{array}$$

(35)

Since the polylogarithm ${\mathrm{Li}}_{-k}\left(z\right)$, $0\le k<\infty$, is a bounded rational function in z when $0\le z<1$, then for $ϵ>0$, the function ${\mathrm{Li}}_{-k}\left(p^{-ϵ}\right)$ is also bounded. Thus, ${\mathcal{R}}_4\left(f;N,p^m,k\right)$ converges to zero and, by (30) and (31), the limit (29) holds.

Extend this result on the entire set of nonnegative real numbers $b_1$. In order to do that, we make use of the $\mathsf{S}\mathsf{Q}$ theorem and start with trivial inequalities,

$$\begin{array}{c}\hfill {\left(1-{\displaystyle \frac{r}{{\log }_{p}N}}\right)}^{⌊b_1+1⌋}\le {\left(1-{\displaystyle \frac{r}{{\log }_{p}N}}\right)}^{b_1}\le {\left(1-{\displaystyle \frac{r}{{\log }_{p}N}}\right)}^{⌊b_1⌋}, 0\le r\le ⌊{\log }_{p}N⌋,\end{array}$$

(36)

which, due to (25), implies the following relations:

$$\begin{array}{ccc}& & {\mathcal{J}}_1\left(f;p^m,{\log }_{p}N,⌊b_1+1⌋\right)\le {\mathcal{J}}_1\left(f;p^m,{\log }_{p}N,b_1\right)\le {\mathcal{J}}_1\left(f;p^m,{\log }_{p}N,⌊b_1⌋\right), \hfill \\ & & \mathrm{where}{\mathcal{J}}_1\left(f;p^m,M,b\right)=\sum _{r=0}^{⌊M⌋}\left|L_r\left(f;p^m\right)\right|p^{-a_{1}r}{\left(1-{\displaystyle \frac{r}{M}}\right)}^b, b\ge 0.\hfill \end{array}$$

(37)

From the proof of convergence of ${\mathcal{R}}_1\left(f;N,p^m\right)$ with nonnegative integer degrees $b_1$ and by (37) and by the $\mathsf{S}\mathsf{Q}$ theorem, it follows that ${\mathcal{R}}_1\left(f;N,p^m\right)$ converges with real $b_1\ge 0$. □

Lemma 1 can be applied to the totient functions $f\left(k\right)=\varphi \left(k\right),\psi \left(k\right)$ and their inverse $1/f\left(k\right)$ with functions $L_r\left(f;p^m\right)$ and $L_r\left(f^{-1};p^m\right)$ calculated in (18).

Lemma 2.

Let a function $f\left(k\right)\in \mathbb{B}\left\{N^{a_1}{\left(\log N\right)}^{b_1};G_2\left(N\right)\right\}$, $a_1>0$, $b_1<0$ be given and let there exist two numbers $\mathcal{K}>0$ and $\gamma <a_1$ and an integer $r_{*}\ge 0$ such that $|L_r\left(f;p^m\right)|\le \mathcal{K}{p}^{\gamma r}$ for all $r\ge r_{*}$. Then, (29) holds.

Proof. 

Consider ${\mathcal{R}}_1\left(f;N,p^m\right)$ given in (25) when $a_1>0$, $b_1\in {\mathbb{Z}}_{-}$, i.e., $b_1$ is a negative integer. To avoid its divergence at $r={\log }_{p}N$, we use representation (20) in Remark 2,

$$\begin{array}{c}\hfill {\mathcal{R}}_1\left(f;N,p^m\right)=\sum _{r=0}^{⌊{\log }_{p}N⌋-1}{\displaystyle \frac{L_r\left(f;p^m\right)}{p^{a_{1}r}}}{\left(1-{\displaystyle \frac{r}{{\log }_{p}N}}\right)}^{-|b_1|}+{\displaystyle \frac{A}{\mathcal{F}}}{\displaystyle \frac{{\left({\log }_{p}N\right)}^{|b_1|}}{N^{a_1}}},\end{array}$$

(38)

where $A=L_{\overline{r}}\left(f;p^m\right)F[f;N_{\overline{r}},1]<\infty$ and $\overline{r}=⌊{\log }_{p}N⌋$ were defined in (20).

Estimate ${\mathcal{R}}_1\left(f;N,p^m\right)$ when $|L_r\left(f;p^m\right)|<\mathcal{K}{p}^{\gamma r}$, $\mathcal{K}>0$ and $\gamma <a_1$. Note that the last term in (38) does not contribute to the asymptotics of ${\mathcal{R}}_1\left(f;N,p^m\right)$ when $N\to \infty$ and can therefore be omitted in what follows. We use an identity

$$\begin{array}{c}\hfill {(1-x)}^{-b}=1+x\sum _{k=1}^b{(1-x)}^{-k},\hspace{1em}b\in {\mathbb{Z}}_{+},\end{array}$$

(39)

and represent (38) as follows:

$$\begin{array}{c}\hfill {\mathcal{R}}_1\left(f;N,p^m\right)\simeq \sum _{r=0}^{⌊{\log }_{p}N⌋-1}{\displaystyle \frac{L_r\left(f;p^m\right)}{p^{a_{1}r}}}+\sum _{k=1}^{|b_1|}{\mathcal{R}}_5\left(f;N,p^m,k\right),\mathrm{where}\end{array}$$

(40)

$$\begin{array}{c}\hfill {\mathcal{R}}_5\left(f;N,p^m,k\right)={\displaystyle \frac{1}{{\log }_{p}N}}\sum _{r=0}^{⌊{\log }_{p}N⌋-1}{\displaystyle \frac{L_r\left(f;p^m\right)}{p^{a_{1}r}}}r{\left(1-{\displaystyle \frac{r}{{\log }_{p}N}}\right)}^{-k}.\end{array}$$

(41)

Note that the following inequality holds:

$$\begin{array}{c}\hfill {\left(1-{\displaystyle \frac{r}{{\log }_{p}N}}\right)}^{-1}\le 1+r if 0\le r<⌊{\log }_{p}N⌋.\end{array}$$

(42)

Substitute (42) into (41) and keep in mind that due to the prefactor ${\left({\log }_{p}N\right)}^{-1}$ in the r.h.s. of Equation (41) the same convergence of ${\mathcal{R}}_5\left(f;N,p^m,k\right)$ holds at intervals $\left(r_{*},⌊{\log }_{p}N⌋-1\right)$ and $\left(0,⌊{\log }_{p}N⌋-1\right)$ (see discussion in the proof of Lemma 1). Then, we arrive at the following estimate:

$$\begin{array}{c}\hfill |{\mathcal{R}}_5\left(f;N,p^m,k\right)|\le {\displaystyle \frac{\mathcal{K}}{{\log }_{p}N}}\sum _{r=0}^{⌊{\log }_{p}N⌋-1}{\displaystyle \frac{r{(1+r)}^k}{p^{ϵr}}}={\displaystyle \frac{\mathcal{K}}{{\log }_{p}N}}\sum _{j=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right)\sum _{r=0}^{⌊{\log }_{p}N⌋-1}{\displaystyle \frac{r^{j+1}}{p^{ϵr}}}.\end{array}$$

The rest of the proof follows by applying the same arguments of asymptotics of the Hurwitz–Lerch zeta function, as it was performed in Lemma 1,

$$\begin{array}{c}\hfill |{\mathcal{R}}_5\left(f;N,p^m,k\right)|\le \mathcal{K}\sum _{j=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right)\left[{\displaystyle \frac{{\mathrm{Li}}_{-(j+1)}\left(p^{-ϵ}\right)}{{\log }_{p}N}}-{\displaystyle \frac{1}{p^{ϵ}-1}}{\displaystyle \frac{{\left({\log }_{p}N\right)}^j}{N^{ϵ}}}\right].\end{array}$$

(43)

By comparing the r.h.s. in (43) and (35), we conclude that ${\mathcal{R}}_5\left(f;N,p^m,k\right)$ is convergent to zero when $N\to \infty$. Thus, by (40) the limit (29) holds for $b_1\in {\mathbb{Z}}_{-}$.

We extend this result by the $\mathsf{S}\mathsf{Q}$ theorem on all negative real $b_1$. This can be performed by inequality (37) for another function ${\mathcal{J}}_2\left(f;p^m,M,b\right)$,

$$\begin{array}{c}\hfill {\mathcal{J}}_2\left(f;p^m,{\log }_{p}N,⌊b_1+1⌋\right)\le {\mathcal{J}}_2\left(f;p^m,{\log }_{p}N,b_1\right)\le {\mathcal{J}}_2\left(f;p^m,{\log }_{p}N,⌊b_1⌋\right),\end{array}$$

(44)

where

$$\begin{array}{c}\hfill {\mathcal{J}}_2\left(f;p^m,M,b\right)=\sum _{r=0}^{⌊M⌋-1}{\displaystyle \frac{|L_r\left(f;p^m\right)|}{p^{a_{1}r}}}{\left(1-{\displaystyle \frac{r}{M}}\right)}^b, b<0.\end{array}$$

From the proof of the convergence of ${\mathcal{R}}_1\left(f;N,p^m\right)$ with negative integer degrees $b_1$ and by (44) and by the $\mathsf{S}\mathsf{Q}$ Theorem, it follows that ${\mathcal{R}}_1\left(f;N,p^m\right)$ converges with real $b_1<0$. □

2.2. Convergence of ${\mathcal{R}}_2\left(f;N,p^m\right)$

In this section, we consider the convergence of ${\mathcal{R}}_2\left(f;N,p^m\right)$, $N\to \infty$, defined in (22) and responsible for the contribution of the error term to $R_{\infty }\left(f;p^m\right)$.

Lemma 3.

Let a function $f\left(k\right)\in \mathbb{B}\left\{N^{a_1}{\left(\log N\right)}^{b_1};N^{-a_2}{\left(\log N\right)}^{-b_2}\right\}$, $b_1\ge b_2$, be given and let there exist two numbers $\mathcal{K}>0$, $\gamma <a_1$ and integer $r_{*}\ge 0$ such that $|L_r\left(f;p^m\right)|\le \mathcal{K}{p}^{\gamma r}$ for all $r\ge r_{*}$, then

$$\begin{array}{c}\hfill \underset{N\to \infty }{\lim }{\mathcal{R}}_2\left(f;N,p^m\right)=0.\end{array}$$

(45)

Proof. 

Denote $b_1-b_2=e$ and make use of a simple inequality, ${(1-x)}^e\le 1$ when $0\le x\le 1$, $e\ge 0$. Formula (26) can be rewritten as follows:

$$\begin{array}{c}\hfill \left|{\mathcal{R}}_2\left(f;N,p^m\right)\right|\le {\displaystyle \frac{\mathcal{C}}{N^{a_2}{\left(\log N\right)}^{b_2}}}\sum _{r=0}^{⌊{\log }_{p}N⌋}{\displaystyle \frac{\left|L_r\left(f;p^m\right)\right|}{p^{(a_1-a_2)r}}}\le {\displaystyle \frac{\mathcal{C}\mathcal{K}}{N^{a_2}{\left(\log N\right)}^{b_2}}}\sum _{r=0}^{⌊{\log }_{p}N⌋}{p}^{\nu r},\end{array}$$

(46)

where $\nu =a_2-a_1+\gamma$. Keep in mind that due to the prefactor $N^{-a_2}{\left(\log N\right)}^{-b_2}$ in (46), the same convergence of the r.h.s. in (46) holds at intervals $\left(r_{*},⌊{\log }_{p}N⌋\right)$ and $\left(0,⌊{\log }_{p}N⌋\right)$ (see discussion in proof of Lemma 1).

Further calculations depend on the sign of $\nu$. If $\nu <0$, then

$$\begin{array}{c}\hfill \gamma <a_1-a_2, \left|{\mathcal{R}}_2\left(f;N,p^m\right)\right|\le {\displaystyle \frac{\mathcal{C}\mathcal{K}}{1-p^{\nu }}}{\displaystyle \frac{N^{-a_2}}{{\left(\log N\right)}^{b_2}}}.\end{array}$$

(47)

If $\nu =0$, then

$$\begin{array}{c}\hfill \gamma =a_1-a_2, \left|{\mathcal{R}}_2\left(f;N,p^m\right)\right|\le \mathcal{C}\mathcal{K}{\displaystyle \frac{N^{\gamma -a_1}}{{\left(\log N\right)}^{b_2-1}}}.\end{array}$$

(48)

Finally, if $\nu >0$, then

$$\begin{array}{c}\hfill \gamma >a_1-a_2, \left|{\mathcal{R}}_2\left(f;N,p^m\right)\right|\le {\displaystyle \frac{\mathcal{C}\mathcal{K}}{N^{a_2}{\left(\log N\right)}^{b_2}}}{\displaystyle \frac{p^{\nu \left(⌊{\log }_{p}N⌋+1\right)}}{p^{\nu }-1}}\simeq {\displaystyle \frac{\mathcal{C}\mathcal{K}}{1-p^{-\nu }}}{\displaystyle \frac{N^{\gamma -a_1}}{{\left(\log N\right)}^{b_2}}}. \end{array}$$

(49)

Require now that all r.h.s. in (47), (48) and (49) converge to zero when $N\to \infty$. Regarding the first case (47), this always holds because by (8) if $a_2=0$, then $b_2>0$, and if $a_2>0$, then $b_2\ge 0$, so the r.h.s. in (47) is a decreasing function. So, this leads to the requirement, $\gamma <a_1-a_2$. In two other cases, we have the following necessary conditions:

$$\begin{array}{ccc}& & a_1-a_2=\gamma \le a_1 \mathrm{if} b_2>1, \mathrm{and} a_1-a_2=\gamma <a_1 \mathrm{if} -\infty <b_2<\infty ,\hfill \end{array}$$

(50)

$$\begin{array}{ccc}& & a_1-a_2<\gamma \le a_1 \mathrm{if} b_2>0, \mathrm{and} a_1-a_2<\gamma <a_1 \mathrm{if} -\infty <b_2<\infty .\hfill \end{array}$$

(51)

Summarizing the necessary conditions (50), (51) and $\gamma <a_1-a_2$, we conclude that the numerical series ${\mathcal{R}}_2\left(f;N,p^m\right)$ is convergent to zero absolutely when $N\to \infty$ and irrespectively to the sign of $b_2$ if $\gamma <a_1$. This proves Formula (45). □

Lemma 4.

Let a function $f\left(k\right)\in \mathbb{B}\left\{N^{a_1}{\left(\log N\right)}^{b_1};N^{-a_2}{\left(\log N\right)}^{-b_2}\right\}$, $b_1<b_2$ be given and let there exist two numbers $\mathcal{K}>0$ and $\gamma <a_1$ and an integer $r_{*}\ge 0$ such that $|L_r\left(f;p^m\right)|\le \mathcal{K}{p}^{\gamma r}$ for all $r\ge r_{*}$, then (45) holds.

Proof. 

Consider ${\mathcal{R}}_3\left(f;N,p^m\right)$ given in (27) and, according to Remark 2, rewrite it as follows:

$$\begin{array}{c}\hfill {\mathcal{R}}_3\left(f;N,p^m\right)=\sum _{r=0}^{⌊{\log }_{p}N⌋-1}{\displaystyle \frac{\left|L_r\left(f;p^m\right)\right|}{p^{(a_1-a_2)r}}}{\left(1-{\displaystyle \frac{r}{{\log }_{p}N}}\right)}^{-|b_2-b_1|}+{\displaystyle \frac{A}{\mathcal{F}}}{\displaystyle \frac{N^{-a_1}}{{\left({\log }_{p}N\right)}^{b_1}}},\end{array}$$

(52)

where the upper bound in the sum is taken in order to avoid its divergence and A is defined in (38); for more details, see the proof of Lemma 2, formula (38).

The last term in (52) does not contribute to the asymptotics of ${\mathcal{R}}_2\left(f;N,p^m\right)$ when $N\to \infty$ and can be skipped. Indeed, if $a_1\ge 0$, $b_1>0$, it converges to zero when $N\to \infty$; in the case $a_1=b_1=0$, according to (8), we have $a_2>0$ or $a_2=0$, $b_2>0$ that again makes it irrelevant due to the prefactor $N^{-a_2}{\left({\log }_{p}N\right)}^{-b_2}$ in formula (26) for ${\mathcal{R}}_2\left(f;N,p^m\right)$. Apply an inequality (42) in the range $0\le r\le ⌊{\log }_{p}N⌋-1$,

$$\begin{array}{c}\hfill {\left(1-{\displaystyle \frac{r}{{\log }_{p}N}}\right)}^{-(b_2-b_1)}\le {\left({\log }_{p}N\right)}^{b_2-b_1},\end{array}$$

and substitute it into (52)

$$\begin{array}{c}\hfill {\mathcal{R}}_3\left(f;N,p^m\right)\le {\left({\log }_{p}N\right)}^{b_2-b_1}\sum _{r=0}^{⌊{\log }_{p}N⌋-1}{\displaystyle \frac{\left|L_r\left(f;p^m\right)\right|}{p^{r(a_1-a_2)}}}.\end{array}$$

(53)

If $a_2>0$, we apply to (53) the constraints on $L_r\left(f;p^m\right)$ and substitute the result into (68),

$$\begin{array}{c}\hfill {\mathcal{R}}_2\left(f;N,p^m\right)\le {\displaystyle \frac{\mathcal{C}\mathcal{K}}{N^{a_2}{\left({\log }_{p}N\right)}^{b_1}}}\sum _{r=0}^{⌊{\log }_{p}N⌋-1}{p}^{\nu r}, \nu =a_2-a_1+\gamma .\end{array}$$

(54)

By comparing (54) with (46) from Lemma 3, we obtain the following, according to (47)–(49):

$$\begin{array}{ccc}\hfill \left|{\mathcal{R}}_2\left(f;N,p^m\right)\right|& \le & {\displaystyle \frac{\mathcal{C}\mathcal{K}}{p^{\nu }-1}}{\displaystyle \frac{N^{\gamma -a_1}}{{\left({\log }_{c}N\right)}^{b_1}}}, \nu >0,\hfill \end{array}$$

(55)

$$\begin{array}{ccc}\hfill \left|{\mathcal{R}}_2\left(f;N,p^m\right)\right|& \le & \mathcal{C}\mathcal{K}{\displaystyle \frac{N^{-a_2}}{{\left({\log }_{c}N\right)}^{b_1-1}}}, \nu =0.\hfill \end{array}$$

(56)

$$\begin{array}{ccc}\hfill \left|{\mathcal{R}}_2\left(f;N,p^m\right)\right|& \le & {\displaystyle \frac{\mathcal{C}\mathcal{K}}{1-p^{\nu }}}{\displaystyle \frac{N^{-a_2}}{{\left({\log }_{c}N\right)}^{b_1}}}, \nu <0.\hfill \end{array}$$

(57)

Thus, by (55)–(57) and inequalities $\gamma <a_1$, $a_2>0$, the series ${\mathcal{R}}_2\left(f;N,p^m\right)$ is convergent to zero when $N\to \infty$, irrespective of the value of $b_1$.

Consider another case $a_2=0$ and compare Formulas (38) and (52) for $b_1<0$ and $b_1<b_2$, respectively. A difference in degrees, $b_1-b_2$ and $b_1$, does not break the main result of Lemma 2: only the first leading term in (40) is survived when $N\to \infty$. When we apply it to (52) and make use of constraint on $L_r\left(f;p^m\right)$, we obtain

$$\begin{array}{c}\hfill {\mathcal{R}}_3\left(f;N,p^m\right)\stackrel{N\to \infty }{\simeq }\sum _{r=0}^{⌊{\log }_{p}N⌋-1}{\displaystyle \frac{L_r\left(f;p^m\right)}{p^{a_{1}r}}}\le \mathcal{K}\sum _{r=0}^{\infty }{p}^{-ϵr}={\displaystyle \frac{\mathcal{K}}{1-p^{-ϵ}}},\end{array}$$

where $ϵ=a_1-\gamma >0$. Substituting the last estimate into (68), we obtain

$$\begin{array}{c}\hfill \left|{\mathcal{R}}_2\left(f;N,p^m\right)\right|\le {\displaystyle \frac{\mathcal{C}\mathcal{K}}{1-p^{-ϵ}}}·{\left({\log }_{p}N\right)}^{-b_2}.\end{array}$$

(58)

Recall that by (8), the degrees of the error term satisfy: if $a_2=0$ then $b_2>0$. Thus, by (58), the series ${\mathcal{R}}_2\left(f;N,p^m\right)$ converges to zero when $N\to \infty$, which proves Lemma. □

Let us summarize the results of four Lemmas 1–4 on the convergence of the numerical series ${\mathcal{R}}_1\left(f;N,p^m\right)$ and ${\mathcal{R}}_2\left(f;N,p^m\right)$.

Theorem 1.

Let a function $f\left(k\right)\in \mathbb{B}\left\{G_1\left(N\right);G_2\left(N\right)\right\}$ be given with $a_i$ and $b_i$ satisfying (8), and let there exist two numbers $\mathcal{K}>0$, $\gamma <a_1$ and an integer $r_{*}\ge 0$ such that $|L_r\left(f;p^m\right)|\le \mathcal{K}{p}^{\gamma r}$ for all $r\ge r_{*}$. Then,

$$\begin{array}{c}\hfill R_{\infty }\left(f,p^m\right)=\sum _{r=0}^{\infty }{\displaystyle \frac{L_r\left(f;p^m\right)}{p^{a_{1}r}}}.\end{array}$$

(59)

Proof. 

By Lemmas 1 and 2, if there exist two numbers $\mathcal{K}>0$ and $\gamma <a_1$ and an integer $r_{*}\ge 0$ such that $|L_r\left(f;p^m\right)|\le \mathcal{K}{p}^{\gamma r}$ for all $r\ge r_{*}$, then ${\mathcal{R}}_1\left(f;N,p^m\right)$ converges to ${\sum }_{r=0}^{\infty }{L}_r\left(f;p^m\right)p^{-a_{1}r}$ in the whole range (8) of varying parameters $a_1,b_1$ and $a_2,b_2$. On the other hand, according to Lemmas 3 and 4 under the same sufficient conditions, the numerical series ${\mathcal{R}}_2\left(f;N,p^m\right)$ converges to zero in the same range (8) of varying parameters. Then, according to (24), we arrive at (59). □

There are a few questions about the convergence of ${\mathcal{R}}_1\left(f;N,p^m\right)$ and ${\mathcal{R}}_2\left(f;N,p^m\right)$ that have been left open beyond the scope of Theorem 1. First, this is a problem of necessary convergence conditions that requires further discussion. Another question arises in view of convergence to zero of ${\mathcal{R}}_2\left(f;N,p^m\right)$: it can happen that for some $f\left(k\right)$, both renormalization functions ${\mathcal{R}}_1\left(f;N,p^m\right)$ and ${\mathcal{R}}_2\left(f;N,p^m\right)$ are convergent to nonzero values, while $L_r\left(f;p^m\right)$ is satisfying less strong conditions than those given in Theorem 1. Thus, the following question has been left open.

Question 1.

Does the error term contribute to the renormalization function $R_{\infty }\left(f;p^m\right)$, and which multiplicative arithmetic functions $f\left(k\right)$ can give an affirmative answer?

2.3. Rational Representation of Renormalization Function

In Section 2.1 and Section 2.2, we find requirements sufficient to make ${\mathcal{R}}_1\left(f;N,p^m\right)$ converge and ${\mathcal{R}}_2\left(f;N,p^m\right)$ equal to zero. These conditions are presented through the characteristic functions $L_r\left(f;p^m\right)$ given recursively in (16). Their straightforward formula (17) look cumbersome and lead in certain cases to rather complex expressions, e.g., (19). That is why, in this section, we give another representation for $R_{\infty }\left(f;p^m\right)$ avoiding the use of $L_r\left(f;p^m\right)$.

Substituting (16) into (59), we obtain an infinite series for $R_{\infty }\left(f;p^m\right)$,

$$\begin{array}{ccc}\hfill R_{\infty }\left(f;p^m\right)& =& f\left(p^m\right)+{\displaystyle \frac{f\left(p^{m+1}\right)-L_0\left(f;p^m\right)f\left(p\right)}{p^{a_1}}}+\hfill \\ & & {\displaystyle \frac{f\left(p^{m+2}\right)-L_1\left(f;p^m\right)f\left(p\right)-L_0\left(f;p^m\right)f\left(p^2\right)}{p^{2a_1}}}+\hfill \\ & & {\displaystyle \frac{f\left(p^{m+3}\right)-L_2\left(f;p^m\right)f\left(p\right)-L_1\left(f;p^m\right)f\left(p^2\right)-L_0\left(f;p^m\right)f\left(p^3\right)}{p^{2a_1}}}+\dots .\hfill \end{array}$$

Recasting the terms in the last expression, we obtain

$$\begin{array}{c}\hfill R_{\infty }\left(f;p^m\right)=\sum _{r=0}^{\infty }{\displaystyle \frac{f\left(p^{m+r}\right)}{p^{a_{1}r}}}-{\displaystyle \frac{f\left(p\right)}{p^{a_1}}}\sum _{r=0}^{\infty }{\displaystyle \frac{L_r\left(f;p^m\right)}{p^{a_{1}r}}}-{\displaystyle \frac{f\left(p^2\right)}{p^{2a_1}}}\sum _{r=0}^{\infty }{\displaystyle \frac{L_r\left(f;p^m\right)}{p^{a_{1}r}}}-\dots .\end{array}$$

(60)

Introduce two numerical series $\mathbf{U}\left(f;p^m\right)$ and $\mathbf{V}\left(f;p\right)$

$$\begin{array}{c}\hfill \mathbf{U}\left(f;p^m\right)=\sum _{r=0}^{\infty }{\displaystyle \frac{f\left(p^{r+m}\right)}{p^{a_{1}r}}},\mathbf{V}\left(f;p\right)=\sum _{r=1}^{\infty }{\displaystyle \frac{f\left(p^r\right)}{p^{a_{1}r}}},\end{array}$$

(61)

and assume that they are convergent. Then, by a comparison of (60) and (59), we obtain

$$\begin{array}{c}\hfill R_{\infty }\left(f;p^m\right)={\displaystyle \frac{\mathbf{U}\left(f;p^m\right)}{1+\mathbf{V}\left(f;p\right)}}.\end{array}$$

(62)

if the denominator in (62) does not vanish. In short, this can be written as follows: $\mathbf{V}\left(f;p\right)+1={\sum }_{r=0}^{\infty }f\left(p^r\right)p^{-a_{1}r}$. However, we prefer to stay with (62); otherwise, one can make a mistake in the calculations, e.g., $\varphi \left(p^r\right)=p^{r-1}(p-1)$, $r\ge 1$, and $\varphi \left(p^0\right)=1$, but $\varphi \left(p^0\right)\ne p^{-1}(p-1)$.

Formula (62) gives a rational representation of the renormalization function $R_{\infty }\left(f;p^m\right)$ that is free of intermediate calculations of $L_r\left(f;p^m\right)$. What can be said about the convergence of $R_{\infty }\left(f;p^m\right)$ in terms of $f\left(p^r\right)$?

Theorem 2.

Let a function $f\left(k\right)\in \mathbb{B}\left\{G_1\left(N\right);G_2\left(N\right)\right\}$ be given and let there exist two numbers ${\mathcal{K}}_1>0$ and ${\gamma }_1<a_1$ and an integer $r_{*}\ge 0$ such that $|f\left(p^r\right)|\le {\mathcal{K}}_1{p}^{{\gamma }_{1}r}$ for all $r\ge r_{*}$. Then, $R_{\infty }\left(f;p^m\right)$ is convergent in accordance with (62) if $\mathbf{V}\left(f;p\right)+1\ne 0$.

Proof. 

Constraints $|f\left(p^r\right)|\le {\mathcal{K}}_1{p}^{{\gamma }_{1}r}$, ${\mathcal{K}}_1>0$ and ${\gamma }_1<a_1$, for all $r\ge r_{*}$ imply the convergence of $\mathbf{U}\left(f;p^m\right)$ and $\mathbf{V}\left(f;p\right)$, which follows by their absolute convergence.

$$\begin{array}{ccc}& & \left|\mathbf{U}\left(f;p^m\right)\right|\le {\mathcal{K}}_1{p}^{{\gamma }_{1}m}\sum _{r=0}^{\infty }{p}^{({\gamma }_1-a_1)r}={\displaystyle \frac{{\mathcal{K}}_1{p}^{{\gamma }_{1}m}}{1-p^{{\gamma }_1-a_1}}},\hfill \\ & & \left|\mathbf{V}\left(f;p\right)\right|\le {\mathcal{K}}_1\sum _{r=0}^{\infty }{p}^{({\gamma }_1-a_1)r}={\displaystyle \frac{{\mathcal{K}}_1}{1-p^{{\gamma }_1-a_1}}}.\hfill \end{array}$$

Thus, if the denominator in (62) does not vanish, $\mathbf{V}\left(f;p\right)+1\ne 0$, then $R_{\infty }\left(f;p^m\right)$ is convergent in accordance with (62). □

Table 3 presents different multiplicative functions and their corresponding parameters $a_1$, ${\gamma }_1$, ${\mathcal{K}}_1$, and $r_{*}$. All functions satisfy the constraints of Theorem 2 for $p\ge 2$.

Table 3. Multiplicative functions and their corresponding parameters $a_1$, ${\gamma }_1$, ${\mathcal{K}}_1$, and $r_{*}$.

$\mathit{f}\left(\mathit{k}\right)$	$\mathit{\varphi }\left(\mathit{k}\right)$	$1/\mathit{\varphi }\left(\mathit{k}\right)$	$\mathit{\psi }\left(\mathit{k}\right)$	$1/\mathit{\psi }\left(\mathit{k}\right)$	${\mathit{J}}_{\mathit{n}}\left(\mathit{k}\right)$	${\mathit{\mu }}^2\left(\mathit{k}\right)$	$2^{\mathit{\omega }\left(\mathit{k}\right)}$	$3^{\mathit{\omega }\left(\mathit{k}\right)}$
$a_1$	2	0	2	0	$n+1$	1	1	1
${\gamma }_1$	1	$-1$	1	$-1$	n	0	0	0
${\mathcal{K}}_1$	$(p-1)/p$	$p/(p-1)$	$(p+1)/p$	$p/(p+1)$	$(p^n-1)/p^n$	1	2	3
$r_{*}$	1	1	1	1	1	0	0	0

Regarding the convergence of $\mathbf{U}\left(f;p^m\right)$ and $\mathbf{V}\left(f;p\right)$, defined in (61), we present an example that shows that the renormalization function $R_{\infty }\left(f;p^m\right)$ can exist even when both $\mathbf{U}\left(f;p^m\right)$ and $\mathbf{V}\left(f;p\right)$ are divergent. Consider $f\left(k\right)=2^{\mathrm{\Omega }\left(k\right)}$, where $2^{\mathrm{\Omega }\left(p^r\right)}=2^r$, and obtain

$$\begin{array}{ccc}& & a_1={\gamma }_1=1, {\mathcal{K}}_1=1,\hspace{1em}{L}_0\left(2^{\mathrm{\Omega }\left(k\right)};p^m\right)=2^m,\hspace{1em}{L}_r\left(2^{\mathrm{\Omega }\left(k\right)};p^m\right)=0,\hspace{1em}r\ge 1,\hfill \\ \hfill \\ & & \mathbf{U}\left(f;2^m\right)\stackrel{N\to \infty }{\simeq }\infty ,\hspace{1em}\mathbf{V}\left(f;2\right)\stackrel{N\to \infty }{\simeq }\infty ,\hspace{1em}{R}_{\infty }\left(2^{\mathrm{\Omega }\left(k\right)},2^m\right)=2^m.\hfill \end{array}$$

Here, the conditions of Theorem 1 are satisfied for all $r\ge 1$, and only the first term is left non-zero in series (59). Both $\mathbf{U}\left(f;2^m\right)$ and $\mathbf{V}\left(f;2\right)$ are divergent, e.g., $\mathbf{V}\left(f;2\right)={\sum }_{r=1}^{\infty }1$, and, therefore, Theorem 2 cannot be applied. In other words, Theorem 1 has a much wider area of application than Theorem 2. In the following sections, we use both theorems.

3. Multiplicativity of Renormalization Function with Complex Scaling

In this section, we study the renormalization of a summatory function when the summation variable k is scaled by a product ${\prod }_{i=1}^n{p}_i^{m_i}$ with n distinct primes $p_i\ge 2$. Consider $F\left\{f;N,{\prod }_{i=1}^n{p}_i^{m_i}\right\}$ and find its governing functional Equation (64).

First, write the relationship between two summatory functions with two different summands, $f\left(k_1{p}_n^{m_n}\right)$ and $f\left(k_1\right)$, where $k_1=k{\prod }_{i=1}^{n-1}{p}_i^{m_i}$. It is similar to that given in (15) and follows from the latter by replacing $k\to k_1$, i.e.,

$$\begin{array}{c}\hfill F\left\{f;N,\prod _{i=1}^n{p}_i^{m_i}\right\}=\sum _{r_n=0}^{p_n^{r_n}\le N}{L}_{r_n}\left(f;p_n^{m_n}\right)F\left\{f⌊{\displaystyle \frac{N}{p_n^{r_n}}}⌋,\prod _{i=1}^{n-1}{p}_i^{m_i}\right\}.\end{array}$$

(63)

Next, repeat this procedure to reduce the scale by $p_{n-1}^{r_{n-1}}$ for the summatory function that appeared in the r.h.s. of (63) and substitute it again into (63),

$$\begin{array}{c}\hfill F\left\{f;N,\prod _{i=1}^n{p}_i^{m_i}\right\}=\sum _{r_{n-1},r_n=0}^{p_{n-1}^{n-1}{p}_n^{r_n}\le N}{L}_{r_n}\left(f;p_n^{m_n}\right)L_{r_{n-1}}\left(f;p_{n-1}^{m_{n-1}}\right)F\left\{f;⌊{\displaystyle \frac{N}{p_{n-1}^{r_{n-1}}{p}_n^{r_n}}}⌋,\prod _{i=1}^{n-2}{p}_i^{m_i}\right\}.\end{array}$$

Continue to reduce the scales in a consecutive way for the next summatories and obtain, finally,

$$\begin{array}{c}\hfill F\left\{f;N,\prod _{i=1}^n{p}_i^{m_i}\right\}=\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{r_i}\le N}\left(\prod _{i=1}{L}_{r_i}\left(f;p_i^{m_i}\right)\right)F\left\{f;⌊{\displaystyle \frac{N}{{\prod }_{i=1}^n{p}_i^{r_i}}}⌋,1\right\}.\end{array}$$

(64)

Define new renormalization functions,

$$\begin{array}{ccc}& & R\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)={\displaystyle \frac{F\left\{f;N,{\prod }_{i=1}^n{p}_i^{m_i}\right\}}{F\{f;N,1\}}},\hfill \\ & & R_{\infty }\left(f;\prod _{i=1}^n{p}_i^{m_i}\right):=\underset{N\to \infty }{\lim }R\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right),\hfill \end{array}$$

(65)

and find their representations through the characteristic functions $L_r\left(f;p_i^{m_i}\right)$, $i=1,\dots ,n$, and the degrees $a_1,b_1$ and $a_2,b_2$ of the leading asymptotics $G_1\left(N\right)$ and $G_2\left(N\right)$, respectively. Following an approach developed in Section 2, represent $R\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i}\right)$ as a sum

$$\begin{array}{c}\hfill R\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)={\mathcal{R}}_1\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)+{\mathcal{R}}_2\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right),\end{array}$$

(66)

where ${\mathcal{R}}_j\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i}\right)$, $j=1,2$, are analogous to those given in (22) and (23). We substitute $G_1\left(N\right)$ and $G_2\left(N\right)$, given in (8), and obtain formulas similar to those given in (25)–(27). Here they are,

$$\begin{array}{c}\hfill {\mathcal{R}}_1\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)=\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{r_i}\le N}\prod _{i=1}^n\left({\displaystyle \frac{L_{r_i}\left(f;p_i^{m_i}\right)}{p_i^{a_1{r}_i}}}\right){\left(1-{\displaystyle \frac{1}{{\log }_{c}N}}\sum _{i=1}^n{r}_i{\log }_c{p}_i\right)}^{b_1},\end{array}$$

(67)

where a base c is chosen in such a way that $2\le c<\min \{p_1,\dots ,p_n\}$ so that the upper summation bound ${\prod }_{i=1}^n{p}_i^{m_i}\le N$ is correspondent to inequality, ${\sum }_{j=1}^n{r}_i{\log }_c{p}_i\le {\log }_{c}N$. Regarding ${\mathcal{R}}_2\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i}\right)$, we have an upper bound

$$\begin{array}{c}\hfill \left|{\mathcal{R}}_2\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)\right|\le {\displaystyle \frac{\mathcal{C}}{N^{a_2}{\left({\log }_{c}N\right)}^{b_2}}}{\mathcal{R}}_3\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right),\end{array}$$

(68)

where

$$\begin{array}{c}\hfill {\mathcal{R}}_3\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)=\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{r_i}\le N}\prod _{i=1}^n\left({\displaystyle \frac{\left|L_{r_i}\left(f;p_i^{m_i}\right)\right|}{p_i^{r_i(a_1-a_2)}}}\right){\left(1-{\displaystyle \frac{1}{{\log }_{c}N}}\sum _{i=1}^n{r}_i{\log }_c{p}_i\right)}^{b_1-b_2}. \end{array}$$

(69)

Keeping in mind $\mathsf{S}\mathsf{Q}$ theorem and its usage in Section 2.1 and Section 2.2, we assume throughout this Section $b_1,b_2\in \mathbb{Z}$. The extension to non-integers $b_1$ and $b_2$ is trivial and can be done using Lemmas 1 and 2, and therefore will be omitted. In the next sections, we prove several statements about ${\mathcal{R}}_1\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i}\right)$ and ${\mathcal{R}}_2\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i}\right)$ that are analogous to Lemmas 1–4 in Section 2. In this regard, it is important to use the same sufficient conditions that were used in these lemmas.

3.1. Convergence of ${\mathcal{R}}_1\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i}\right)$

Lemma 5.

Let a function $f\left(k\right)\in \mathbb{B}\left\{N^{a_1}{\left(\log N\right)}^{b_1};G_2\left(N\right)\right\}$, $a_1\ge 0$, $b_1\in {\mathbb{Z}}_{+}\cup \left\{0\right\}$, be given and let there exist two numbers $\mathcal{K}>0$ and $\gamma <a_1$ and an integer $r_{*}\ge 0$ such that $|L_r\left(f;p^m\right)|\le \mathcal{K}{p}^{\gamma r}$ for all $r\ge r_{*}$. Then,

$$\begin{array}{c}\hfill \underset{N\to \infty }{\lim }{\mathcal{R}}_1\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)=\prod _{i=1}^n\left(\sum _{r_i=0}^{\infty }{\displaystyle \frac{L_{r_i}\left(f;p_i^{m_i}\right)}{p_i^{a_1{r}_i}}}\right).\end{array}$$

(70)

Proof. 

Exponentiating the binomial in (67), we obtain

$$\begin{array}{ccc}& & {\mathcal{R}}_1\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)=\hfill \\ & & \sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{r_i}\le N}\left(\prod _{i=1}^n{\displaystyle \frac{L_{r_i}\left(f;p_i^{m_i}\right)}{p_i^{a_1{r}_i}}}\right)+\sum _{k=1}^{b_1}{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{b_1}{k}}\right){\mathcal{R}}_6\left(f;N,\prod _{i=1}^n{p}_i^{m_i},k\right), \hfill \end{array}$$

(71)

where

$$\begin{array}{c}\hfill {\mathcal{R}}_6\left(f;N,\prod _{i=1}^n{p}_i^{m_i},k\right)={\displaystyle \frac{1}{{\left({\log }_{c}N\right)}^k}}\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{r_i}\le N}\left(\prod _{i=1}^n{\displaystyle \frac{L_{r_i}\left(f;p_i^{m_i}\right)}{p_i^{a_1{r}_i}}}\right){\left(\sum _{i=1}^n{r}_i{\log }_c{p}_i\right)}^k.\end{array}$$

Find an estimate for ${\mathcal{R}}_6\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i},k\right)$,

$$\begin{array}{ccc}& & \left|{\mathcal{R}}_6\left(f;N,\prod _{i=1}^n{p}_i^{m_i},k\right)\right|\le \hfill \\ & & {\mathcal{K}}^n\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1,\dots ,k_n=0}{k_1+\dots +k_n=k}}}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{k_1,\dots ,k_n}}\right)\prod _{i=1}^n{\left({\displaystyle \frac{{\log }_c{p}_i}{{\log }_{c}N}}\right)}^{k_i}\hspace{1em}\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{r_i}\le N}\left(\prod _{i=1}^n{\displaystyle \frac{r_i^{k_i}}{p_i^{ϵr_i}}}\right),\hfill \end{array}$$

where $ϵ=a_1-\gamma >0$. One more inequality reads

$$\begin{array}{c}\hfill \sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{r_i}\le N}\left(\prod _{i=1}^n{\displaystyle \frac{r_i^{k_i}}{p_i^{ϵr_i}}}\right)\le \prod _{i=1}^n\left(\sum _{r_i=0}^{p_i^{r_i}\le N}{\displaystyle \frac{r_i^{k_i}}{p_i^{ϵr_i}}}\right)=\prod _{i=1}^{n}T(p_i,k_i,ϵ,{\log }_{p_i}N),\end{array}$$

(72)

where $T(p,k,ϵ,M)$ is defined in (33). Combining the two last inequalities together, we obtain

$$\begin{array}{c}\hfill \left|{\mathcal{R}}_6\left(f;N,\prod _{i=1}^n{p}_i^{m_i},k\right)\right|\le {\mathcal{K}}^n\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1,\dots ,k_n=0}{k_1+\dots +k_n=k}}}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{k_1,\dots ,k_n}}\right)\prod _{i=1}^n{\displaystyle \frac{T(p_i,k_i,ϵ,{\log }_{p_i}N)}{{\left({\log }_{p_i}N\right)}^{k_i}}}.\end{array}$$

(73)

Substituting the asymptotics (35) of $T(p_i,k_i,ϵ,{\log }_{p_i}N)$ into (73), we obtain

$$\begin{array}{c}\hfill \left|{\mathcal{R}}_6\left(f;N,\prod _{i=1}^n{p}_i^{m_i},k\right)\right|\le {\mathcal{K}}^n\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1,\dots ,k_n=0}{k_1+\dots +k_n=k}}}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{k_1,\dots ,k_n}}\right)\prod _{i=1}^n\left({\displaystyle \frac{{\mathrm{Li}}_{-k_i}\left(p_i^{-ϵ}\right)}{{\left({\log }_{p_i}N\right)}^{k_i}}}-{\displaystyle \frac{N^{-ϵ}}{p_i^{ϵ}-1}}\right).\end{array}$$

(74)

Repeating the final remarks in proof of Lemma 1 on the asymptotics of the polylogarithm ${\mathrm{Li}}_s\left(z\right)$, we conclude that ${\mathcal{R}}_6\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i},k\right)$ is convergent to zero when $N\to \infty$. Then, keeping in mind the representation (71) for ${\mathcal{R}}_1\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i}\right)$ and running the upper bound of the summation to infinity, we conclude that the limit (70) holds. □

Lemma 6.

Let a function $f\left(k\right)\in \mathbb{B}\left\{N^{a_1}{\left(\log N\right)}^{b_1};G_2\left(N\right)\right\}$, $a_1>0$, $b_1\in {\mathbb{Z}}_{-}$, be given and let there exist two numbers $\mathcal{K}>0$ and $\gamma <a_1$ and an integer $r_{*}\ge 0$ such that $|L_r\left(f;p^m\right)|\le \mathcal{K}{p}^{\gamma r}$ for all $r\ge r_{*}$. Then, (70) is satisfied.

Proof. 

Here, we follow the keyline in the proof of Lemma 2 and, according to (20) in Remark 2, start with representation of ${\mathcal{R}}_1\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i}\right)$, avoiding its divergence at ${\prod }_{i=1}^n{p}_i^{m_i}=N$,

$$\begin{array}{c}\hfill {\mathcal{R}}_1\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)=\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{m_i}\le N-1}\left(\prod _{i=1}^n{\displaystyle \frac{L_{r_i}\left(f;p_i^{m_i}\right)}{p_i^{a_1{r}_i}}}\right){\left(1-{\displaystyle \frac{1}{{\log }_{c}N}}\sum _{i=1}^n{r}_i{\log }_c{p}_i\right)}^{-|b_1|}. \end{array}$$

(75)

Making use of identity (39), we obtain

$$\begin{array}{c}\hfill {\mathcal{R}}_1\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)=\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{m_i}\le N-1}\left(\prod _{i=1}^n{\displaystyle \frac{L_{r_i}\left(f;p_i^{m_i}\right)}{p_i^{a_1{r}_i}}}\right)+\sum _{k=1}^{|b_1|}{\mathcal{R}}_7\left(f;N,\prod _{i=1}^n{p}_i^{m_i},k\right), \end{array}$$

(76)

where $\tilde{r}={\sum }_{i=1}^n{r}_i{\log }_c{p}_i$ and

$$\begin{array}{c}\hfill {\mathcal{R}}_7\left(f;N,\prod _{i=1}^n{p}_i^{m_i},k\right)={\displaystyle \frac{1}{{\log }_{c}N}}\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{m_i}\le N-1}\left(\prod _{i=1}^n{\displaystyle \frac{L_{r_i}\left(f;p_i^{m_i}\right)}{p_i^{a_1{r}_i}}}\right)\tilde{r}{\left(1-{\displaystyle \frac{\tilde{r}}{{\log }_{c}N}}\right)}^{-k}.\end{array}$$

(77)

Making use of inequality (42) and constraint, imposed on $L_r\left(f;p^m\right)$

$$\begin{array}{c}\hfill {\left(1-{\displaystyle \frac{\tilde{r}}{{\log }_{c}N}}\right)}^{-k}\le {(1+\tilde{r})}^k, \left|L_r\left(f;p^m\right)\right|\le \mathcal{K}{p}^{\gamma r},\end{array}$$

exponentiate the binomial ${(1+\tilde{r})}^k$ in (77) and obtain

$$\begin{array}{ccc}& & \left|{\mathcal{R}}_7\left(f;N,\prod _{i=1}^n{p}_i^{m_i},k\right)\right|\le \hfill \\ & & {\displaystyle \frac{{\mathcal{K}}^n}{{\log }_{c}N}}\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{m_i}\le N-1}{\displaystyle \frac{\tilde{r}{(1+\tilde{r})}^k}{{\prod }_{i=1}^n{p}_i^{ϵr_i}}}={\displaystyle \frac{{\mathcal{K}}^n}{{\log }_{c}N}}\sum _{j=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right)\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{m_i}\le N-1}{\displaystyle \frac{{\tilde{r}}^{j+1}}{p_1^{ϵr_1}{p}_2^{ϵr_2}}},\hfill \end{array}$$

where $ϵ=a_1-\gamma$. Exponentiating the binomial ${\left({\sum }_{i=1}^n{r}_i{\log }_c{p}_i\right)}^{j+1}$ in the last expression, we obtain

$$\begin{array}{ccc}& & \left|{\mathcal{R}}_7\left(f;N,p_1^{m_1}{p}_2^{m_2},k\right)\right|\le \hfill \\ & & {\displaystyle \frac{{\mathcal{K}}^n}{{\log }_{c}N}}\sum _{j=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right)\sum _{{\displaystyle \genfrac{}{}{0pt}{}{j_1,\dots ,j_n=0}{j_1+\dots +j_n=j+1}}}^{j+1}\left({\displaystyle \genfrac{}{}{0pt}{}{j+1}{j_1,\dots ,j_n}}\right)\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{m_i}\le N-1}\prod _{i=1}^n{\displaystyle \frac{r_i^{j_i}}{p_i^{ϵr_i}}}{\left({\log }_c{p}_i\right)}^{j_i}.\hfill \end{array}$$

The asymptotic behavior in N of the last expression is completely determined by its inner sum in $r_i$ with respect to its prefactor ${\left({\log }_{c}N\right)}^{-1}$. This behavior can be studied following the corresponding part in (72) of the proof in Lemma 1,

$$\begin{array}{c}\hfill \sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{m_i}\le N-1}\prod _{i=1}{\displaystyle \frac{r_i^{j_i}}{p_i^{ϵr_i}}}{\left({\log }_c{p}_i\right)}^{j_i}\stackrel{N\to \infty }{\le }\prod _{i=1}\left({\mathrm{Li}}_{-j_i}\left(p_i^{-ϵ}\right)-{\displaystyle \frac{N^{-ϵ}{\left({\log }_{p_i}N\right)}^{j_i}}{1-p_i^{-ϵ}}}\right){\left({\log }_c{p}_i\right)}^{j_i}.\end{array}$$

(78)

Keeping in mind the prefactor ${\left({\log }_{c}N\right)}^{-1}$ and the last asymptotics (78), the upper bound for ${\mathcal{R}}_7\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i},k\right)$ can be performed in an infinitely small way, i.e., it is convergent to zero when $N\to \infty$.

Using the representation (76) for ${\mathcal{R}}_1\left(f;N,p_1^{m_1}{p}_2^{m_2}\right)$ and running the upper bound of summation to infinity, we conclude that the limit (70) holds. □

3.2. Convergence of ${\mathcal{R}}_2\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i}\right)$

Lemma 7.

Let a function $f\left(k\right)\in \mathbb{B}\left\{N^{a_1}{\left(\log N\right)}^{b_1};N^{-a_2}{\left(\log N\right)}^{-b_2}\right\}$, $b_1\ge b_2$, be given and let there exist two numbers $\mathcal{K}>0$ and $\gamma <a_1$ and an integer $r_{*}\ge 0$ such that $|L_r\left(f;p^m\right)|\le \mathcal{K}{p}^{\gamma r}$ for all $r\ge r_{*}$, then

$$\begin{array}{c}\hfill \underset{N\to \infty }{\lim }{\mathcal{R}}_2\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)=0.\end{array}$$

(79)

Proof. 

Denote $b_1-b_2=e\in {\mathbb{Z}}_{+}\cup \left\{0\right\}$ and make use of an inequality ${(1-x)}^e\le 1$ when $0\le x\le 1$, $e\ge 0$. Substituting constraints on $L_r\left(f;p^m\right)$ into (68), we obtain

$$\begin{array}{c}\hfill \left|{\mathcal{R}}_2\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)\right|\le {\displaystyle \frac{\mathcal{C}{\mathcal{K}}^n{\mathcal{I}}_n\left(\nu \right)}{N^{a_2}{\left({\log }_{p}N\right)}^{b_2}}}, \mathrm{where} {\mathcal{I}}_n\left(\nu \right)=\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{r_i}\le N}\prod _{i=1}^n{p}_i^{\nu r_i}.\end{array}$$

(80)

and $\nu =a_2-a_1+\gamma$. Focus on the sum ${\mathcal{I}}_n\left(\nu \right)$ and estimate it in three cases, $\nu =0$, $\nu >0$ and $\nu <0$.

Let $\nu =0$, i.e., $a_2=a_1-\gamma$, then ${\mathcal{I}}_n\left(0\right)$ accounts for a number of integral points (vertices with integer coordinates) in the n-dim simplex, or corner of the n-dim cube, defined as follows:

$$\begin{array}{c}\hfill {\Delta }_n=:\left\{r_1,\dots ,r_n\in {\mathbb{Z}}_{+}\cup \left\{0\right\}|\sum _{i=1}^n{r}_i{\log }_c{p}_i\le {\log }_{c}N\right\}.\end{array}$$

The simplex ${\Delta }_n$ has one orthogonal corner and edge sizes ${\log }_{c}N/{\log }_c{p}_i$ along the ith axis. The number ${\mathcal{I}}_n\left(0\right)$ is described by the Ehrhart polynomial [50] and, when $N\to \infty$, it has a leading term coinciding with simplex volume,

$$\begin{array}{c}\hfill {\mathcal{I}}_n\left(0\right)=\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{r_i}\le N}\simeq {\displaystyle \frac{{\left({\log }_{c}N\right)}^n}{n!{\prod }_{i=1}{\log }_c{p}_i}}.\end{array}$$

(81)

Substituting (81) into (80), we obtain

$$\begin{array}{c}\hfill \left|{\mathcal{R}}_2\left(f;N,\prod _{i=1}^n{p}_i^{r_i}\right)\right|\le {\displaystyle \frac{\mathcal{C}{\mathcal{K}}^n}{n!{\prod }_{i=1}{\log }_c{p}_i}}{\displaystyle \frac{N^{\gamma -a_1}}{{\left({\log }_{c}N\right)}^{b_2-n}}}.\end{array}$$

(82)

By (82) and inequality $\gamma <a_1$, we conclude that ${\mathcal{R}}_2\left(f;N,p_1^{m_1}{p}_2^{m_2}\right)$ is convergent to zero when $N\to \infty$ irrespectively to the value of $b_2$.

Consider the case $\nu >0$ and estimate ${\mathcal{I}}_n\left(\nu \right)$ and ${\mathcal{R}}_2\left(f;N,{\prod }_{i=1}^n{p}_i^{r_i}\right)$,

$$\begin{array}{c}\hfill {\mathcal{I}}_n\left(\nu \right)\le N^{\nu }{\mathcal{I}}_n\left(0\right), \to \left|{\mathcal{R}}_2\left(f;N,\prod _{i=1}^n{p}_i^{r_i}\right)\right|\le {\displaystyle \frac{\mathcal{C}{\mathcal{K}}^n}{n!{\prod }_{i=1}{\log }_c{p}_i}}{\displaystyle \frac{N^{\gamma -a_1}}{{\left({\log }_{c}N\right)}^{b_2-n}}}.\end{array}$$

(83)

By (83), the term ${\mathcal{R}}_2\left(f;N,{\prod }_{i=1}^n{p}_i^{r_i}\right)$ is also convergent to zero when $N\to \infty$.

Finally, consider the case $\nu <0$. According to (80), we have ${\mathcal{I}}_n\left(\nu \right)<{\mathcal{I}}_n\left(0\right)$, and therefore,

$$\begin{array}{c}\hfill \left|{\mathcal{R}}_2\left(f;N,\prod _{i=1}^n{p}_i^{r_i}\right)\right|\le {\displaystyle \frac{\mathcal{C}{\mathcal{K}}^n}{n!{\prod }_{i=1}{\log }_c{p}_i}}{\displaystyle \frac{N^{-a_2}}{{\left({\log }_{c}N\right)}^{b_2-n}}}.\end{array}$$

(84)

Thus, ${\mathcal{R}}_2\left(f;N,{\prod }_{i=1}^n{p}_i^{r_i}\right)$ is convergent to zero due to (84) and constraints (8) on degrees $a_2$ and $b_2$. Summarizing (82)–(84), we complete the proof of Lemma. □

In Lemmas 8 and 9, we consider two different cases, $a_2>0$ and $a_2=0$, separately.

Lemma 8.

Let a function $f\left(k\right)\in \mathbb{B}\left\{N^{a_1}{\left(\log N\right)}^{b_1};N^{-a_2}{\left(\log N\right)}^{-b_2}\right\}$, $b_1<b_2$, $a_2>0$ be given and let there exist two numbers $\mathcal{K}>0$ and $\gamma <a_1$ and an integer $r_{*}\ge 0$ such that $|L_r\left(f;p^m\right)|\le \mathcal{K}{p}^{\gamma r}$ for all $r\ge r_{*}$. Then, the limit (79) is satisfied.

Proof. 

According to (20) in Remark 2, represent ${\mathcal{R}}_3\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i}\right)$ avoiding its divergence at ${\prod }_{i=1}^n{p}_i^{m_i}=N$,

$$\begin{array}{c}\hfill {\mathcal{R}}_3\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)=\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{m_i}\le N-1}\left(\prod _{i=1}^n{\displaystyle \frac{L_{r_i}\left(f;p_i^{m_i}\right)}{p_i^{r_i(a_1-a_2)}}}\right){\left(1-{\displaystyle \frac{\tilde{r}}{{\log }_{c}N}}\right)}^{-(b_2-b_1)},\end{array}$$

(85)

and make use of inequality (42) in the range $0\le \tilde{r}\le ⌊{\log }_{p}N⌋-1$,

$$\begin{array}{c}\hfill {\left(1-{\displaystyle \frac{\tilde{r}}{{\log }_{c}N}}\right)}^{-(b_2-b_1)}\le {\left({\log }_{c}N\right)}^{b_2-b_1}.\end{array}$$

(86)

Substituting (86) into (85) for ${\mathcal{R}}_3\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i}\right)$, we obtain

$$\begin{array}{c}\hfill {\mathcal{R}}_3\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)\le {\left({\log }_{c}N\right)}^{b_2-b_1}\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{m_i}\le N-1}\prod _{i=1}^n{\displaystyle \frac{\left|L_{r_i}\left(f;p_i^{m_i}\right)\right|}{p_i^{r_1(a_1-a_2)}}}.\end{array}$$

(87)

Apply to (87) the constraints on $L_r\left(f;p^m\right)$ and substitute the result into (68),

$$\begin{array}{c}\hfill {\mathcal{R}}_2\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)\le {\displaystyle \frac{\mathcal{C}{\mathcal{K}}^n}{N^{a_2}{\left({\log }_{c}N\right)}^{b_1}}}\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{m_i}\le N-1}\prod _{i=1}^n{p}_i^{\nu r_i}, \nu =a_2-a_1+\gamma .\end{array}$$

(88)

By comparing (88) with (80) from Lemma 7, we obtain according to (82), (83), and (84)

$$\begin{array}{ccc}\hfill \left|{\mathcal{R}}_2\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)\right|& \le & {\displaystyle \frac{\mathcal{C}{\mathcal{K}}^n}{n!{\prod }_{i=1}{\log }_c{p}_i}}{\displaystyle \frac{N^{\gamma -a_1}}{{\left({\log }_{c}N\right)}^{b_1-n}}}, \nu \ge 0,\hfill \end{array}$$

(89)

$$\begin{array}{ccc}\hfill \left|{\mathcal{R}}_2\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)\right|& \le & {\displaystyle \frac{\mathcal{C}{\mathcal{K}}^n}{n!{\prod }_{i=1}{\log }_c{p}_i}}{\displaystyle \frac{N^{-a_2}}{{\left({\log }_{c}N\right)}^{b_1-n}}}, \nu <0.\hfill \end{array}$$

(90)

Thus, by (89) and inequality $\gamma <a_1$, a series ${\mathcal{R}}_2\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i}\right)$ is convergent to zero when $N\to \infty$ irrespectively to the value of $b_1$. The same conclusion (79) about the convergence of this series holds due to (90) when $a_2>0$. □

Lemma 9.

Let a function $f\left(k\right)\in \mathbb{B}\left\{N^{a_1}{\left(\log N\right)}^{b_1};{\left(\log N\right)}^{-b_2}\right\}$, $b_1<b_2$, $a_2=0$ be given and let there exist two numbers $\mathcal{K}>0$ and $\gamma <a_1$ and an integer $r_{*}\ge 0$ such that $|L_r\left(f;p^m\right)|\le \mathcal{K}{p}^{\gamma r}$ for all $r\ge r_{*}$. Then, the limit (79) is satisfied.

Proof. 

This case has to be treated more precisely than that in Lemma 8. Rewrite (85)

$$\begin{array}{c}\hfill {\mathcal{R}}_3\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)=\sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{m_i}\le N-1}\prod _{i=1}^n{\displaystyle \frac{\left|L_{r_i}\left(f;p_i^{m_i}\right)\right|}{p_i^{a_1{r}_i}}}{\left(1-{\displaystyle \frac{\tilde{r}}{{\log }_{c}N}}\right)}^{-(b_2-b_1)},\end{array}$$

(91)

and compare it with expression (75) when $b_1<0$. Changing the powers of $b_1$ to $b_2-b_1$ does not violate the main result of Lemma 6: Only the first leading term in (76) is preserved as $N\to \infty$. When we apply it to (91) and make use of the constraint on $L_r\left(f;p^m\right)$, we obtain

$$\begin{array}{ccc}& & {\mathcal{R}}_3\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)\stackrel{N\to \infty }{=}\hfill \\ & & \sum _{r_1,\dots ,r_n=0}^{{\prod }_{i=1}^n{p}_i^{m_i}\le N-1}\prod _{i=1}^n{\displaystyle \frac{L_{r_i}\left(f;p_i^{m_i}\right)}{p_i^{a_1{r}_i}}}\le {\mathcal{K}}^n\prod _{i=1}^n\left(\sum _{r_i=0}^{\infty }{p}_i^{-ϵr_i}\right)={\displaystyle \frac{{\mathcal{K}}^n}{{\prod }_{i=1}^n\left(1-p_i^{-ϵ}\right)}},\hfill \end{array}$$

where $ϵ=a_1-\gamma >0$. Substituting the last estimate into (68), we obtain

$$\begin{array}{c}\hfill \left|{\mathcal{R}}_2\left(f;N,\prod _{i=1}^n{p}_i^{m_i}\right)\right|\le {\displaystyle \frac{\mathcal{C}{\mathcal{K}}^n}{{\prod }_{i=1}^n\left(1-p_i^{-ϵ}\right)}}·{\displaystyle \frac{1}{{\left({\log }_{p}N\right)}^{b_2}}}.\end{array}$$

(92)

Recall that by (8), the degrees of the error term satisfy the following condition: if $a_2=0$, then $b_2>0$. Thus, by (92) the series ${\mathcal{R}}_2\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i}\right)$ converges to zero when $N\to \infty$, which proves the lemma. □

We combine Lemmas 5–9 on the convergence of ${\mathcal{R}}_j\left(f;N,{\prod }_{i=1}^n{p}_i^{m_i}\right)$, $j=1,2$, and, according to (66), arrive at an analog of Theorem 1 in the case of scaling by ${\prod }_{i=1}^n{p}_i^{m_i}$.

Theorem 3.

Let a function $f\left(k\right)\in \mathbb{B}\left\{G_1\left(N\right);G_2\left(N\right)\right\}$ be given with $a_i$ and $b_i$ satisfying (8) and let there exist two numbers $\mathcal{K}>0$ and $\gamma <a_1$ and an integer $r_{*}\ge 0$ such that $|L_r\left(f;p^m\right)|\le \mathcal{K}{p}^{\gamma r}$ for all $r\ge r_{*}$. Then,

$$\begin{array}{c}\hfill R_{\infty }\left(f;\prod _{i=1}^n{p}_i^{m_i}\right)=\prod _{i=1}^n\left(\sum _{r_i=0}^{\infty }{\displaystyle \frac{L_{r_i}\left(f;p_i^{m_i}\right)}{p_i^{a_1{r}_i}}}\right).\end{array}$$

(93)

Combining Theorems 1 and 3, we arrive at an important consequence that manifests the multiplicative property of the renormalization function.

Corollary 1.

Under the conditions of Theorem 3, the following holds:

$$\begin{array}{c}\hfill R_{\infty }\left(f;\prod _{i=1}^n{p}_i^{m_i}\right)=\prod _{i=1}^n{R}_{\infty }\left(f;p_i^{m_i}\right).\end{array}$$

(94)

3.3. Asymptotics of Summatory Functions ${\sum }_{k_1,k_2\le N}f\left(k_1{k}_2\right)$

In this section, we calculate the summatory function $\mathrm{\Phi }[f;N,1]={\sum }_{k_1,k_2\le N}f\left(k_1{k}_2\right)$ and find its asymptotics by applying Corollary 1. According to the definition of a summatory function, we obtain

$$\begin{array}{ccc}\hfill \mathrm{\Phi }\{f;N,1\}& =& \sum _{k\le N}f\left(k\right)+\sum _{k\le N}f\left(2k\right)+\dots +\sum _{k\le N}f\left(Nk\right)=\hfill \\ & & \sum _{k\le N}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{m_{i,j}=0}{n_j=1}}}f\left(k\prod _{i=1}^{n_j}{p}_i^{m_{i,j}}\right)=\sum _{{\displaystyle \genfrac{}{}{0pt}{}{m_{i,j}=0}{n_j=1}}}\sum _{k\le N}f\left(k\prod _{i=1}^{n_j}{p}_i^{m_{i,j}}\right),\hfill \end{array}$$

where indices i, j, $m_{i,j}$, and $n_j$ account for all primes $p_i$ such that ${\prod }_{i=1}^{n_j}{p}_i^{m_{i,j}}\le N$. Thus, according to the definition of the summatory function with scaled summation variable, we obtain

$$\begin{array}{c}\hfill \mathrm{\Phi }\{f;N,1\}=\sum _{{\displaystyle \genfrac{}{}{0pt}{}{m_{i,j}=0}{n_j=1}}}F\left\{f;N,\prod _{i=1}^{n_j}{p}_i^{m_{i,j}}\right\}, {\displaystyle \frac{\mathrm{\Phi }\{f;N,1\}}{F\{f;N,1\}}}=\sum _{{\displaystyle \genfrac{}{}{0pt}{}{m_{i,j}=0}{n_j=1}}}R\left(f;N,\prod _{i=1}^{n_j}{p}_i^{m_{i,j}}\right),\end{array}$$

(95)

where ${\prod }_{i=1}^{n_j}{p}_i^{m_{i,j}}\le N$. Consider asymptotics (omitting the error terms) of three summatory functions when $N\to \infty$,

$$\begin{array}{ccc}& & F\{f;N,1\}\stackrel{N\to \infty }{=}\mathcal{F}{G}_1\left(N\right),\mathrm{\Phi }\{f;N,1\}\stackrel{N\to \infty }{=}{\mathcal{G}}_1\left(f\right){\Gamma }_1\left(N\right),\hfill \\ & & \sum _{{\displaystyle \genfrac{}{}{0pt}{}{m_{i,j}=0}{n_j=1}}}R\left(f;N,\prod _{i=1}^{n_j}{p}_i^{m_{i,j}}\right)\stackrel{N\to \infty }{=}{\mathcal{G}}_2\left(f\right){\Gamma }_2\left(N\right).\hfill \end{array}$$

(96)

Combining (96) and the 2nd formula in (95), we obtain

$$\begin{array}{c}\hfill {\Gamma }_1\left(N\right)=G_1\left(N\right)·{\Gamma }_2\left(N\right), {\mathcal{G}}_1\left(f\right)=\mathcal{F}·{\mathcal{G}}_2\left(f\right).\end{array}$$

(97)

Calculation of ${\Gamma }_2\left(N\right)$ and ${\mathcal{G}}_2\left(f\right)$ is a difficult numerical task. Consider a special case if ${\mathcal{G}}_2\left(f\right)$ can be given in a closed form, namely if ${\Gamma }_2\left(N\right)=N^0$, that is, ${\Gamma }_1\left(N\right)=G_1\left(N\right)$. Consider the third asymptotics in (96) and, according to Corollary 1, find its limit when $N\to \infty$,

$$\begin{array}{ccc}\hfill {\mathcal{G}}_2\left(f\right)& =& \underset{N\to \infty }{\lim }\sum _{{\displaystyle \genfrac{}{}{0pt}{}{m_{i,j}=0}{n_j=1}}}^{{\prod }_{i=1}^{n_j}{p}_i^{m_{i,j}}\le N}R\left(f;N,\prod _{i=1}^{n_j}{p}_i^{m_{i,j}}\right)=\underset{{\displaystyle \genfrac{}{}{0pt}{}{N\to \infty }{n_j\to \infty }}}{\lim }\sum _{{\displaystyle \genfrac{}{}{0pt}{}{m_{i,j}=0}{n_j=1}}}^{{\log }_{p_i}N}\prod _{i=1}^{n_j}{R}_{\infty }\left(f;p_i^{m_{i,j}}\right)\hfill \\ & =& \underset{n\to \infty }{\lim }\prod _{i=1}^n\sum _{m=0}^{\infty }{R}_{\infty }\left(f;p_i^m\right)=\prod _{p\ge 2}\sum _{m=0}^{\infty }{R}_{\infty }\left(f;p^m\right).\hfill \end{array}$$

(98)

Another special case comes when $f\left(k\right)$ is a completely multiplicative arithmetic function, i.e., $f\left(k_1{k}_2\right)=f\left(k_1\right)f\left(k_2\right)$. That leads to the following equalities: ${\Gamma }_2\left(N\right)\equiv {\Gamma }_1\left(N\right)$ and ${\mathcal{G}}_2\left(f\right)=\mathcal{F}$. We illustrate this statement and formula (98) in Section 4.2.

4. Renormalization of Dirichlet Series and Others Summatory Functions

In this section, we extend the renormalization approach to summatory functions of more complex structures. They involve the summatory functions with summands given by ${\prod }_{i=1}^n{f}_i\left(k\right)$ and summation variable k scaled for every multiplicative function $f_i$ by $p^{m_i}$, $m_i\ne m_j$. The case of the Dirichlet series is special when $n=2$ and $f_2\left(k\right)=k^{-s}$. We also study the renormalization of summatory functions with summands given by $f\left(k^n\right)$.

4.1. Renormalization of Summatory Functions ${\sum }_{k\le N}{\prod }_{i=1}^n{f}_i\left(k{p}^{m_i}\right)$

Start with a summatory function

$$F\left\{\prod _{i=1}^n{f}_i;N,p^{\mathbf{m}}\right\}=\sum _{k\le N}\prod _{i=1}^n{f}_i\left(k{p}^{m_i}\right),$$

where $p^{\mathbf{m}}$ denotes a tuple $\left\{p^{m_1},\dots ,p^{m_n}\right\}$, and use a standard notation $F\left\{{\prod }_{i=1}^n{f}_i;N,1^{\mathbf{m}}\right\}=F\left\{{\prod }_{i=1}^n{f}_i;N,1\right\}$. Derive for F a functional equation following the approach developed in Section 1.2 and start

$$\begin{array}{ccc}\hfill F\left\{\prod _{i=1}^n{f}_i;N,p^{\mathbf{m}}\right\}& =& \mathsf{F}\left(p^{\mathbf{m}}\right)\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k=1}{p\nmid k}}}^N\prod _{i=1}^n{f}_i\left(k\right)+\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k=p}{p\mid k}}}^N\prod _{i=1}^n{f}_i\left(k{p}^{m_i}\right)\hfill \\ & =& \mathsf{F}\left(p^{\mathbf{m}}\right)\left(\sum _{k=1}^N\prod _{i=1}^n{f}_i\left(k\right)-\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k=p}{p\mid k}}}^N\prod _{i=1}^n{f}_i\left(k\right)\right)+\sum _{l=1}^{N_1}\prod _{i=1}^n{f}_i\left(l{p}^{m_i+1}\right),\hfill \end{array}$$

where $\mathsf{F}\left(p^{\mathbf{m}}\right)={\prod }_{i=1}^n{f}_i\left(p^{m_i}\right)$ and $N_r$ are defined in Section 1.2. Rewrite the last equality

$$\begin{array}{c}\hfill F\left\{\prod _{i=1}^n{f}_i;N,p^{\mathbf{m}}\right\}-F\left\{\prod _{i=1}^n{f}_i;N_1,p^{\mathbf{m}+\mathbf{1}}\right\}=\mathsf{F}\left(p^{\mathbf{m}}\right)\left[F\left\{\prod _{i=1}^n{f}_i;N,1\right\}-F\left\{\prod _{i=1}^n{f}_i;N_1,p\right\}\right],\end{array}$$

which is similar to (13). The corresponding counter-partner for its general version (14) reads,

$$\begin{array}{ccc}& & F\left\{\prod _{i=1}^n{f}_i;N_r,p^{\mathbf{m}}\right\}-F\left\{\prod _{i=1}^n{f}_i;N_{r+1},p^{\mathbf{m}+\mathbf{1}}\right\}=\hfill \\ & & \mathsf{F}\left(p^{\mathbf{m}}\right)\left(F\left\{\prod _{i=1}^n{f}_i;N_r,1\right\}-F\left\{\prod _{i=1}^n{f}_i;N_{r+1},p\right\}\right).\hfill \end{array}$$

Combining the last equations of the running index $0\le r\le ⌊{\log }_{p}N⌋$ together, we arrive at the functional equation for the summatory function,

$$\begin{array}{c}\hfill F\left\{\prod _{i=1}^n{f}_i;N,p^{\mathbf{m}}\right\}=\sum _{r=0}^{⌊{\log }_{p}N⌋}{L}_r\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)F\left\{\prod _{i=1}^n{f}_i;N_r,1\right\},\end{array}$$

(99)

where

$$\begin{array}{c}\hfill L_r\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)=\prod _{i=1}^n{f}_i\left(p^{m_i+r}\right)-\sum _{j=0}^{r-1}{L}_j\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)\prod _{i=1}^n{f}_i\left(p^{r-j}\right).\end{array}$$

(100)

Formulas for the first $L_r\left({\prod }_{i=1}^n{f}_i;p^{\mathbf{m}}\right)$ read

$$\begin{array}{ccc}\hfill L_0\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)& =& \prod _{i=1}^n{f}_i\left(p^{m_i}\right),\hfill \\ \hfill L_1\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)& =& \prod _{i=1}^n{f}_i\left(p^{m_i+1}\right)-\prod _{i=1}^n{f}_i\left(p^{m_i}\right)\prod _{i=1}^n{f}_i\left(p\right),\hfill \\ \hfill L_2\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)& =& \prod _{i=1}^n{f}_i\left(p^{m_i+2}\right)-\prod _{i=1}^n{f}_i\left(p^{m_i}\right)\prod _{i=1}^n{f}_i\left(p^2\right)-\prod _{i=1}^n{f}_i\left(p^{m_i+1}\right)\prod _{i=1}^n{f}_i\left(p\right)+\hfill \\ & & \prod _{i=1}^n{f}_i\left(p^{m_i}\right)\prod _{i=1}^n{f}_i^2\left(p\right), etc\hfill \end{array}$$

(101)

such that for $p=1$ or $m=0$, we have, $L_0\left({\prod }_{i=1}^n{f}_i;1^{\mathbf{m}}\right)=1$ and $L_r\left({\prod }_{i=1}^n{f}_i;1^{\mathbf{m}}\right)=0$, $r\ge 1$.

Find the analog to Formula (18) when $f_i\left(p^m\right)=A_{f_i}{p}^{m-1}$, $m\ge 1$, and $A_{f_i}$ denotes the real constant. By (100) or (101), such formulas for $L_r\left({\prod }_{i=1}^n{f}_i;p^{\mathbf{m}}\right)$ can be calculated by induction,

$$\begin{array}{ccc}\hfill L_r\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)& =& p^{m_1+\dots +m_n-n}{\left(p^n-\prod _{i=1}^n{A}_{f_i}\right)}^r\prod _{i=1}^n{A}_{f_i},\hfill \\ \hfill L_r\left(\prod _{i=1}^n{\displaystyle \frac{1}{f_i}};p^{\mathbf{m}}\right)& =& p^{n-m_1-\dots -m_n}{\left(p^{-n}-\prod _{i=1}^n{A}_{f_i}^{-1}\right)}^r\prod _{i=1}^n{A}_{f_i}^{-1}.\hfill \end{array}$$

E.g., in the case of the Euler $f_1=\varphi \left(k\right)$ and Dedekind $f_2=\psi \left(k\right)$ totient functions, we have

$$\begin{array}{c}\hfill L_r\left(\varphi \psi ;p^{\mathbf{m}}\right)=(p^2-1)p^{m_1+m_2-2}, L_r\left({\displaystyle \frac{1}{\varphi \psi }};p^{\mathbf{m}}\right)={\displaystyle \frac{{(-1)}^r{p}^{2(1-r)-m_1-m_2}}{{(p^2-1)}^{r+1}}}.\end{array}$$

Define new renormalization functions,

$$\begin{array}{c}\hfill R\left(\prod _{i=1}^n{f}_i;N,p^{\mathbf{m}}\right)={\displaystyle \frac{F\left\{{\prod }_{i=1}^n{f}_i;N,p^{\mathbf{m}}\right\}}{F\{{\prod }_{i=1}^n{f}_i;N,1\}}}, R_{\infty }\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)=\underset{N\to \infty }{\lim }R\left(\prod _{i=1}^n{f}_i;N,p^{\mathbf{m}}\right).\end{array}$$

(102)

By comparing formulas (99)–(101) with (15)–(17), respectively, and the definition of (102) with (5), we conclude that all results on the renormalization of summatory functions $F\left\{f;N,p^m\right\}$ in Section 2 can be reproduced for summatory functions $F\left\{{\prod }_{i=1}^n{f}_i;N,1^{\mathbf{m}}\right\}$ with a few necessary alterations.

In Theorem 4, we give (without proof) a sufficient condition for convergence of the asymptotics of the renormalization function $R_{\infty }\left({\prod }_{i=1}^n{f}_i;p^{\mathbf{m}}\right)$. Its proof does not use new ideas and can be given following Lemmas 1–4 for the renormalization function $R_{\infty }\left(f;p^m\right)$. For this reason, we omitted this proof here.

Theorem 4.

Let n multiplicative functions $f_i\left(k\right)$ be given such that $f_i\left(k\right)\in \mathbb{B}\left\{G_1\left(N\right);G_2\left(N\right)\right\}$ satisfies (8). Let there exist two numbers $\mathcal{K}>0$ and $\gamma <a_1$ and an integer $r_{*}\ge 0$ such that $|L_r\left({\prod }_{i=1}^n{f}_i;p^{\mathbf{m}}\right)|\le \mathcal{K}{p}^{\gamma r}$ for all $r\ge r_{*}$. Then,

$$\begin{array}{c}\hfill R_{\infty }\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)=\sum _{r=0}^{\infty }{L}_r\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)p^{-a_{1}r}.\end{array}$$

(103)

To study $R_{\infty }\left({\prod }_{i=1}^n{f}_i;p^{\mathbf{m}}\right)$ in a way similar to the study of $R_{\infty }\left(f;p^m\right)$ in Section 2, we find another representation for $R_{\infty }\left({\prod }_{i=1}^n{f}_i;p^{\mathbf{m}}\right)$ that is different from (103). Substitute (100) into (103) and obtain

$$\begin{array}{c}\hfill R_{\infty }\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)=\prod _{i=1}^n{f}_i\left(p^{m_i}\right)+\left[\prod _{i=1}^n{f}_i\left(p^{m_i+1}\right)-L_0\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)\prod _{i=1}^n{f}_i\left(p\right)\right]p^{-a_1}+\\ \hfill \left[\prod _{i=1}^n{f}_i\left(p^{m_i+2}\right)-L_1\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)\prod _{i=1}^n{f}_i\left(p\right)-L_0\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)\prod _{i=1}^n{f}_i\left(p^2\right)\right]p^{-2a_1}+\dots \end{array}$$

Recasting the terms in the last expression, we obtain

$$\begin{array}{ccc}& & R_{\infty }\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)=\hfill \\ & & \sum _{r=0}^{\infty }{p}^{-a_{1}r}\prod _{i=1}^n{f}_i\left(p^{m_i+r}\right)-\sum _{r=1}^{\infty }{p}^{-a_{1}r}\prod _{i=1}^n{f}_i\left(p^r\right)·\sum _{r=0}^{\infty }{L}_r\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)p^{-a_{1}r}.\hfill \end{array}$$

Thus, by comparison the last expression with Formula (103), we obtain

$$\begin{array}{c}\hfill R_{\infty }\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)={\displaystyle \frac{\mathbf{U}\left({\prod }_{i=1}^n{f}_i;p^{\mathbf{m}}\right)}{1+\mathbf{V}\left({\prod }_{i=1}^n{f}_i;p\right)}},\end{array}$$

(104)

where two numerical series

$$\begin{array}{c}\hfill \mathbf{U}\left(\prod _{i=1}^n{f}_i;p^{\mathbf{m}}\right)=\sum _{r=0}^{\infty }{p}^{-a_{1}r}\prod _{i=1}^n{f}_i\left(p^{m_i+r}\right),\mathbf{V}\left(\prod _{i=1}^n{f}_i;p\right)=\sum _{r=1}^{\infty }{p}^{-a_{1}r}\prod _{i=1}^n{f}_i\left(p^r\right)\end{array}$$

(105)

are assumed to be convergent, and a denominator in (104) does not vanish. Formula (104) gives a rational representation of the renormalization function $R_{\infty }\left({\prod }_{i=1}^n{f}_i;p^{\mathbf{m}}\right)$ free of intermediate calculations of $L_r\left({\prod }_{i=1}^n{f}_i;p^{\mathbf{m}}\right)$.

4.2. Renormalization of the Dirichlet Series ${\sum }_{k=1}^{\infty }f\left(k{p}^m\right)k^{-s}$

The Dirichlet series

$$D\left(f;p^m,s\right)=\sum _{k=1}^{\infty }{\displaystyle \frac{f\left(k{p}^m\right)}{k^s}}$$

with a scaled summation variable is a special case of the summatory function $F\left\{f_1{f}_2;N,p^{\mathbf{m}}\right\}$, discussed in Section 4.1, when $f_1=f\left(k\right)$, $f_2=k^{-s}$, $m_1=m$, $m_2=0$, and $f_1{f}_2\in \mathbb{B}\left\{N^0;G_2\left(N\right)\right\}$, i.e., $a_1=0$.

According to Theorem 4 and Formula (104), if there exist two numbers ${\mathcal{K}}_1>0$ and ${\gamma }_1<0$ and an integer $r_{*}\ge 0$ such that $|f\left(p^r\right)|\le {\mathcal{K}}_1{p}^{(s+{\gamma }_1)r}$ for all $r\ge r_{*}$, then

$$\begin{array}{c}\hfill \mathfrak{D}\left(f;p^m,s\right)={\displaystyle \frac{D\left(f;p^m,s\right)}{D\left(f,s\right)}}=\sum _{r=0}^{\infty }{\displaystyle \frac{f\left(p^{r+m}\right)}{p^{sr}}}{\left(1+\sum _{r=1}^{\infty }{\displaystyle \frac{f\left(p^r\right)}{p^{sr}}}\right)}^{-1}.\end{array}$$

(106)

where $D\left(f,s\right)={\sum }_{k=1}^{\infty }f\left(k\right)k^{-s}$ is a standard Dirichlet series for arithmetic function $f\left(k\right)$. Note that according to the definition (102) of the renormalization function $R_{\infty }\left(f·k^{-s};p^m\right)$, the following equality holds for the Dirichlet series: $\mathfrak{D}\left(f;p^m,s\right)=p^{sm}{R}_{\infty }\left(f·k^{-s};p^m\right)$.

We present four examples of the Dirichlet series $D\left(f;p^m,s\right)$ for the Möbius $\mu \left(k\right)$, Liouville $\lambda \left(k\right)$, Euler $\varphi \left(k\right)$, and divisor ${\sigma }_n\left(k\right)$ functions. Their standard Dirichlet series $D\left(f,s\right)$ converge to the values presented in Table 2.

• Consider two Dirichlet series for the $\mu$-function, $D\left(\mu ,s\right)=1/\zeta \left(s\right)$ and $D\left({\mu }^2,s\right)=\zeta \left(s\right)/\zeta \left(2s\right)$, $s>1$, and calculate their scaled versions. Since $\mu \left(p^{m}k\right)\equiv 0$, $m\ge 2$, we consider here only the case $m=1$ and have ${\mu }^q\left(p^{r+1}\right)={(-1)}^q{\delta }_{0,r}$, $q=1,2$. Then,

  $$\begin{array}{c}\hfill \mathfrak{D}\left(\mu ;p,s\right)=-{\displaystyle \frac{p^s}{p^s-1}},\hspace{1em}\mathfrak{D}\left({\mu }^2;p,s\right)={\displaystyle \frac{p^s}{p^s+1}}.\end{array}$$

  (107)
  Calculate the Dirichlet series $\Delta \left({\mu }^q,s\right)={\sum }_{k_1,k_2=1}^{\infty }{\left(k_1{k}_2\right)}^{-s}{\mu }^q\left(k_1{k}_2\right)$, $q=1,2$, according to (98)

  $$\begin{array}{ccc}& & \Delta \left(\mu ,s\right)={\displaystyle \frac{1}{\zeta \left(s\right)}}{\mathcal{G}}_2\left({\displaystyle \frac{\mu }{k^s}}\right),\hspace{1em}\Delta \left({\mu }^2,s\right)={\displaystyle \frac{\zeta \left(s\right)}{\zeta \left(2s\right)}}{\mathcal{G}}_2\left({\displaystyle \frac{{\mu }^2}{k^s}}\right),\hfill \\ & & {\mathcal{G}}_2\left({\displaystyle \frac{\mu }{k^s}}\right)=\prod _{p\ge 2}\left(1+R_{\infty }\left({\displaystyle \frac{\mu }{k^s}};p\right)\right)=\prod _{p\ge 2}\left(1-{\displaystyle \frac{1}{p^s-1}}\right)<{\displaystyle \frac{1}{\zeta \left(s\right)}},\hfill \\ & & {\mathcal{G}}_2\left({\displaystyle \frac{{\mu }^2}{k^s}}\right)=\prod _{p\ge 2}\left(1+R_{\infty }\left({\displaystyle \frac{{\mu }^2}{k^s}};p\right)\right)=\prod _{p\ge 2}\left(1+{\displaystyle \frac{1}{p^s+1}}\right)<{\displaystyle \frac{\zeta \left(s\right)}{\zeta \left(2s\right)}}.\hfill \end{array}$$

  (108)
  That results in inequalities, $\Delta (\mu ,s)<D^2(\mu ,s)$ and $\Delta \left({\mu }^2,s\right)<D^2\left({\mu }^2,s\right)$. A normalized product for $s=2$ in (108) is known as the Feller–Tornier constant $C_{FT}$ [51],

  $$\begin{array}{c}\hfill {\displaystyle \frac{1}{\zeta \left(2\right)}}\prod _{p\ge 2}\left(1-{\displaystyle \frac{1}{p^2-1}}\right)=\prod _{p\ge 2}\left(1-{\displaystyle \frac{2}{p^2}}\right)=C_{FT}=0.32263.\end{array}$$
• Let us consider the Dirichlet series for the $\lambda$-function, $D\left(\lambda ,s\right)=\zeta \left(2s\right)/\zeta \left(s\right)$, $s>1$, and calculate its scaled version. Keeping in mind $\lambda \left(p^r\right)={(-1)}^r$, we obtain $\mathfrak{D}\left(\lambda ;p^m,s\right)={(-1)}^m$. Calculate the Dirichlet series $\Delta \left(\lambda ,s\right)={\sum }_{k_1,k_2=1}^{\infty }{\left(k_1{k}_2\right)}^{-s}\lambda \left(k_1{k}_2\right)$ in accordance with (98)

  $$\begin{array}{ccc}& & \Delta \left(\lambda ,s\right)={\displaystyle \frac{\zeta \left(2s\right)}{\zeta \left(s\right)}}{\mathcal{G}}_2\left({\displaystyle \frac{\lambda }{k^s}}\right),\hfill \\ & & {\mathcal{G}}_2\left({\displaystyle \frac{\lambda }{k^s}}\right)=\prod _{p\ge 2}\sum _{m=0}^{\infty }{(-1)}^m{p}^{-sm}=\prod _{p\ge 2}{\displaystyle \frac{1}{1+p^{-s}}}={\displaystyle \frac{\zeta \left(2s\right)}{\zeta \left(s\right)}},\hfill \end{array}$$

  (109)

  i.e., $\Delta \left(\lambda ,s\right)=D^2\left(\lambda ,s\right)$ in accordance with the fact that $\lambda \left(k\right)$ is completely multiplicative.
• Consider the Dirichlet series for the $\varphi$-function, $D\left(\varphi ,s\right)=\zeta (s-1)/\zeta \left(s\right)$, $s>2$, and calculate its scaled version. Keeping in mind $\varphi \left(p^r\right)=(p-1)p^{r-1}$, we obtain

  $$\begin{array}{c}\hfill \mathfrak{D}\left(\varphi ;p^m,s\right)={\displaystyle \frac{(p-1)p^{m-1}}{1-p^{-s}}}, m\ge 1.\end{array}$$

  (110)
• Let us consider two Dirichlet series for the ${\sigma }_0$-function, namely, $D\left({\sigma }_0,s\right)={\zeta }^2\left(s\right)$ and $D\left({\sigma }_0^2,s\right)={\zeta }^4\left(s\right)/\zeta \left(2s\right)$, $s\ge 1$, and calculate its scaled version. Keeping in mind ${\sigma }_0\left(p^r\right)=r+1$, we obtain

  $$\begin{array}{ccc}\hfill \mathfrak{D}\left({\sigma }_0;p^m,s\right)& =& (m+1)(1-p^{-s})+p^{-s},m\ge 0,\hfill \\ \hfill \\ \hfill \mathfrak{D}\left({\sigma }_0^2;p^m,s\right)& =& {\displaystyle \frac{{\left((m+1)(1-p^{-s})+p^{-s}\right)}^2+p^{-s}}{1+p^{-s}}}.\hfill \end{array}$$

  (111)

We finish the section with the relationship between characteristic functions for multiplicative arithmetic functions $f\left(k\right)$ and $f_1\left(k\right)=f\left(k\right)·k^{-s}$

$$\begin{array}{c}\hfill L_r\left(f·k^{-s};p^m\right)=L_r\left(f;p^m\right)p^{-(m+r)s},\end{array}$$

(112)

which is followed by (16) and (17) if we substitute the identity $f_1\left(p^r\right)=f\left(p^r\right)p^{-rs}$.

The relation (112) is used in Section 5.3 when calculating the renormalized Dirichlet series for the Ramanujan $\tau$ function.

4.3. Renormalization of Summatory Functions ${\sum }_{k\le N}f\left(k^n{p}^m\right)$

Consider the summatory function $F\left\{f,n;N,p^m\right\}={\sum }_{k\le N}f\left(k^n{p}^m\right)$ and derive its governing functional equation following the approach developed in Section 1.2,

$$\begin{array}{ccc}& & F\left\{f,n;N,p^m\right\}=f\left(p^m\right)\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k=1}{p\nmid k}}}^{N}f\left(k^n\right)+\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k=p}{p\mid k}}}^{N}f\left(p^m{k}^n\right)=\hfill \\ & & f\left(p^m\right)\left(\sum _{k=1}^N-\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k=p}{p\mid k}}}^N\right)f\left(k^n\right)+\sum _{l=1}^{N_1}f\left(p^{m+n}{l}^n\right),\hfill \end{array}$$

which can be rewritten as follows,

$$\begin{array}{c}\hfill F\left\{f,n;N,p^m\right\}-F\left\{f,n;N_1,p^{m+n}\right\}=f\left(p^m\right)\left[F\left\{f,n;N,1\right\}-F\left\{f,n;N_1,p^n\right\}\right].\end{array}$$

(113)

In a general case ($r\ge 1$), Equation (113) has the form

$$\begin{array}{ccc}& & F\left\{f,n;N_r,p^{m+rn}\right\}-F\left\{f,n;N_{r+1},p^{m+(r+1)n}\right\}=\hfill \\ & & f\left(p^{m+rn}\right)\left[F\left\{f,n;N_r,1\right\}-F\left\{f,n;N_{r+1},p^n\right\}\right].\hfill \end{array}$$

Combining the last equations of the running index $0\le r\le ⌊{\log }_{p}N⌋$, we arrive at the functional equation

$$\begin{array}{ccc}\hfill F\left\{f,n;N_r,p^m\right\}& =& \sum _{r=0}^{⌊{\log }_{p}N⌋}{L}_r\left(f,n;p^m\right)F\{f,n;N_r,1\}, \mathrm{where}\hfill \end{array}$$

(114)

$$\begin{array}{ccc}\hfill L_r\left(f,n;p^m\right)& =& f\left(p^{m+rn}\right)-\sum _{j=0}^{r-1}{L}_j\left(f,n;p^m\right)f\left(p^{(r-j)n}\right). \hfill \end{array}$$

(115)

The straightforward calculations of $L_r\left(f;p^m\right)$ give

$$\begin{array}{ccc}& & L_0\left(f,n;p^m\right)=f\left(p^m\right),\hfill \\ & & L_1\left(f,n;p^m\right)=f\left(p^{m+n}\right)-f\left(p^m\right)f\left(p^n\right),\hfill \\ & & L_2\left(f,n;p^m\right)=f\left(p^{m+2n}\right)-f\left(p^{m+n}\right)f\left(p^n\right)-f\left(p^m\right)\left[f\left(p^{2n}\right)-f^2\left(p^n\right)\right].\hfill \end{array}$$

(116)

If $n=1$, then Formulas (114)–(116) are reduced to (15)–(17).

Define new renormalization functions,

$$\begin{array}{c}\hfill R\left(f,n;N,p^m\right)={\displaystyle \frac{F\left\{f,n;N,p^m\right\}}{F\left\{f,n;N,1\right\}}},\hspace{1em}{R}_{\infty }\left(f,n;p^m\right):=\underset{N\to \infty }{\lim }R\left(f,n;N,p^m\right).\end{array}$$

(117)

By comparing formulas (114)–(116) with (15)–(17), respectively, and definition (102) with (117), we conclude that all results on the renormalization of summatory functions $F\left\{f;N,p^m\right\}$ in Section 2 can be reproduced for summatory functions $F\left\{f,n;N,p^m\right\}$ with a few necessary alterations.

In the following, we present (without proof) Theorem 5 on a sufficient condition to converge of asymptotics of the renormalization function $R_{\infty }\left(f,n;p^m\right)$. As in the case of Theorem 4 on the renormalization function $R_{\infty }\left(f_1{f}_2;p^{\mathbf{m}}\right)$, here, the proof of Theorem 5 does not use new ideas and can be given following Lemmas 1–4 for the renormalization function $R_{\infty }\left(f;p^m\right)$. For this reason, we skip it here.

Theorem 5.

Let a function $f\left(k^n\right)\in \mathbb{B}\left\{G_1\left(N\right);G_2\left(N\right)\right\}$ be given and let there exist two numbers ${\mathcal{K}}_1>0$, ${\gamma }_1<a_1$ and an integer $r_{*}\ge 0$ such that $|f\left(p^r\right)|\le {\mathcal{K}}_1{p}^{{\gamma }_{1}r}$ for all $r\ge r_{*}$. Then,

$$\begin{array}{c}\hfill R_{\infty }\left(f,n;p^m\right)=\sum _{r=0}^{\infty }{\displaystyle \frac{L_r\left(f,n;p^m\right)}{p^{a_{1}r}}}.\end{array}$$

(118)

Recasting the terms in the last expression, we obtain

$$\begin{array}{c}\hfill R_{\infty }\left(f,n;p^m\right)=\sum _{r=0}^{\infty }{\displaystyle \frac{f\left(p^{m+nr}\right)}{p^{a_{1}r}}}-\sum _{r=0}^{\infty }{\displaystyle \frac{L_r\left(f,n;p^m\right)}{p^{a_{1}r}}}\left[{\displaystyle \frac{f\left(p^n\right)}{p^{a_1}}}+{\displaystyle \frac{f\left(p^{2n}\right)}{p^{2a_1}}}+{\displaystyle \frac{f\left(p^{3n}\right)}{p^{3a_1}}}+\dots \right].\end{array}$$

Thus, by comparing the last expression with Formula (103), we obtain

$$\begin{array}{c}\hfill R_{\infty }\left(f,n;p^m\right)={\displaystyle \frac{\mathbf{U}\left(f,n;p^m\right)}{1+\mathbf{V}\left(f,n;p\right)}},\end{array}$$

(119)

where two numerical series

$$\begin{array}{c}\hfill \mathbf{U}\left(f,n;p^m\right)=\sum _{r=0}^{\infty }{\displaystyle \frac{f\left(p^{m+nr}\right)}{p^{a_{1}r}}},\mathbf{V}\left(f,n;p\right)=\sum _{r=1}^{\infty }{\displaystyle \frac{f\left(p^{rn}\right)}{p^{a_{1}r}}},\end{array}$$

(120)

are assumed to be convergent, and a denominator in (119) does not vanish.

We apply formulas (119) and (120) to calculate the following Dirichlet series $D\left({\sigma }_0,2,s,p^m\right)={\sum }_{k=1}^{\infty }{k}^{-s}{\sigma }_0\left(k^2{p}^m\right)$, keeping in mind [1] the standard Dirichlet series $D\left({\sigma }_0,2,s,1\right)={\zeta }^3\left(s\right)/\zeta \left(2s\right)$, i.e., $a_1=0$. We reduce our problem as follows:

$$\begin{array}{c}\hfill D\left({\sigma }_0,2,s,p^m\right)=p^{sm/2}\sum _{k=1}^{\infty }{\displaystyle \frac{{\sigma }_0\left(k^2{p}^m\right)}{{\left(k^2{p}^m\right)}^{s/2}}}=p^{sm/2}{R}_{\infty }\left({\displaystyle \frac{{\sigma }_0\left(k\right)}{k^{s/2}}},2;p^m\right){\displaystyle \frac{{\zeta }^3\left(s\right)}{\zeta \left(2s\right)}},\end{array}$$

(121)

and calculate the renormalization function in (121). According to (120), we obtain

$$\begin{array}{ccc}\hfill p^{sm/2}\mathbf{U}\left({\displaystyle \frac{{\sigma }_0\left(k\right)}{k^{s/2}}},2;p^m\right)& =& {\displaystyle \frac{p^s}{p^s-1}}\left(m+1+{\displaystyle \frac{2}{p^s-1}}\right),\hfill \\ \hfill 1+\mathbf{V}\left({\displaystyle \frac{{\sigma }_0\left(k\right)}{k^{s/2}}},2;p\right)& =& {\displaystyle \frac{p^s}{p^s-1}}\left(1+{\displaystyle \frac{2}{p^s-1}}\right).\hfill \end{array}$$

Thus, following (119) and (121), we arrive finally at

$$\begin{array}{c}\hfill D\left({\sigma }_0,2,s,p^m\right)={\displaystyle \frac{(m+1)(p^s-1)+2}{p^s+1}}{\displaystyle \frac{{\zeta }^3\left(s\right)}{\zeta \left(2s\right)}}.\end{array}$$

(122)

5. Renormalization of the Basic Summatory Functions

In this section, we calculate the renormalization function $R_{\infty }\left(f;p^m\right)$ for various summatory functions presented in Table 1 and Table 2. For this purpose, almost all summatory functions are considered by Theorem 2 and the corresponding formulas (60) and (61), based on the calculation of $f\left(p^r\right)$. However, in Section 5.3, we present another approach, which follows Theorem 1, and calculate the characteristic functions $L_r\left(f;p^m\right)$ for the Ramanujan $\tau$ function.

5.1. Renormalization of Summatory Totient Functions

We apply the renormalization approach to summatory totient functions and the Dirichlet series involving the Jordan $J_n\left(k\right)$, Euler $\varphi \left(k\right)$, and Dedekind $\psi \left(k\right)$ functions and their combinations. For the first two functions, we use the technical results given in [52]. When discussing universality classes $\mathbb{B}\{G_1\left(N\right);G_2\left(N\right)\}$, we omit hereafter the error term $G_2\left(N\right)$.

5.1.1. Euler $\varphi \left(k\right)$ Function

Denote the asymptotics of the summatory functions $F\left\{k^u{\varphi }^v;N,1\right\}={\sum }_{k\le N}{k}^u{\varphi }^v\left(k\right)$ in three different ranges of varying parameters $-\infty <u<\infty$ and $v\in \mathbb{Z}$, which is given in [52],

$$F\left\{k^u{\varphi }^v;N,1\right\}\stackrel{N\to \infty }{=}\left\{\begin{array}{ccc}A(u,v)N^{u+v+1}\hfill & \hfill ,& u+v>-1,\hfill \\ \hfill \\ B(u,v)\ln N\hfill & \hfill ,& u+v=-1,\hfill \\ \hfill \\ C(u,v)\hfill & \hfill ,& u+v<-1,\hfill \end{array}\right.$$

such that

$$A(u,0)={\displaystyle \frac{1}{u+1}},\hspace{1em}B(-1,0)=1,\hspace{1em}C(u,0)=\zeta (-u).$$

In the case of arbitrary u and v, many expressions for $A(u,v)$, $B(u,v)$ and $C(u,v)$ can be found in [1,42,43,53,54]. Here, we focus on the renormalization functions that do not specify explicit expressions. According to [52], if $v\ne 0$, then $A(u,v)$, $B(u,v)$, and $C(u,v)$ are bounded from above as follows: if $v\in {\mathbb{Z}}_{+}$, then

$$\begin{array}{c}\hfill 0<A(u,v)\le {\displaystyle \frac{{(u+v+1)}^{-1}}{\zeta (v+1)}}, 0<B(u,v)\le {\displaystyle \frac{1}{\zeta (v+1)}}, 0<C(u,v)\le {\displaystyle \frac{\zeta (-u-v)}{\zeta (-u)}},\end{array}$$

and if $v\in {\mathbb{Z}}_{-}$, then

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{u+v+1}}<& A(u,v)& \le 2^{{\displaystyle \frac{\left|v\right|}{2}}}{\displaystyle \frac{{\mathcal{D}}_{\infty }(v,1)}{u+v+1}},\hspace{1em}{\mathcal{D}}_{\infty }(v,s)=\prod _{r=1}^{\left|v\right|}\zeta \left(s+{\displaystyle \frac{r}{2}}\right),\hfill \\ \hfill 1<& B(u,v)& \le 2^{{\displaystyle \frac{\left|v\right|}{2}}}{\mathcal{D}}_{\infty }(v,1),\hfill \\ \hfill \\ \hfill \zeta (-u-v)<& C(u,v)& \le 2^{{\displaystyle \frac{\left|v\right|}{2}}}{\mathcal{D}}_{\infty }(v,-u-v)\zeta (-u-v).\hfill \end{array}$$

Calculate renormalization functions $R_{\infty }\left(k^u{\varphi }^v;p^m\right)$ according to (60) and (61),

$$\begin{array}{ccc}\hfill R_{\infty }\left(k^u{\varphi }^v;p^m\right)& =& p^{m(u+v)+1}{\displaystyle \frac{{(p-1)}^{v-1}}{p^v+{(p-1)}^{v-1}}}, u+v\ge -1,\hfill \\ \hfill R_{\infty }\left(k^u{\varphi }^v;p^m\right)& =& p^{(m-1)(u+v)}{\displaystyle \frac{{(p-1)}^v}{p^{-u}-p^v+{(p-1)}^v}}, u+v<-1.\hfill \end{array}$$

(123)

By (123) and (124) we have an equality $R_{\infty }\left(k^u{J}_1;p^m\right)=R_{\infty }\left(k^u\varphi ;p^m\right)$ in all ranges of u.

5.1.2. Jordan $J_v\left(k\right)$ and Dedekind $\psi \left(k\right)$ Functions

Regarding the Jordan function $J_v\left(k\right)$, asymptotics of the summatory functions $F\left\{k^u{J}_v;N,1\right\}={\sum }_{k\le N}{k}^u{J}_v\left(k\right)$, $v\in {\mathbb{Z}}_{+}$ can be given in three different ranges of varying parameters [52],

$$\begin{array}{c}\hfill F\left\{k^u{J}_v;N,1\right\}\stackrel{N\to \infty }{\simeq }\left\{\begin{array}{ccc}{\displaystyle \frac{{(u+v+1)}^{-1}}{\zeta (v+1)}}{N}^{u+v+1}\hfill & \hfill ,& u+v>-1,\hfill \\ {\displaystyle \frac{1}{\zeta (v+1)}}lnN\hfill & \hfill ,& u+v=-1,\hfill \\ {\displaystyle \frac{\zeta (-u-v)}{\zeta (-u)}}\hfill & \hfill ,& u+v<-1.\hfill \end{array}\right.\end{array}$$

Calculate their renormalization functions $R_{\infty }\left(k^u{J}_v;p^m\right)$ in accordance with (60) and (61)

$$\begin{array}{ccc}& & {\displaystyle \frac{R_{\infty }\left(k^u{J}_v;p^m\right)}{p^v-1}}={\displaystyle \frac{p^{m(u+v)+1}}{p^{v+1}-1}}, \hspace{1em}u+v\ge -1,\hfill \\ & & {\displaystyle \frac{R_{\infty }\left(k^u{J}_v;p^m\right)}{p^v-1}}={\displaystyle \frac{p^{(m-1)(u+v)}}{p^{-u}-1}}, u+v<-1.\hfill \end{array}$$

(124)

Consider other summatory functions $F\left\{k^u{\psi }^v;N,1\right\}={\sum }_{k\le N}{k}^u{\psi }^v\left(k\right)$ in three different ranges of varying parameters $-\infty <u<\infty$ and $v\in \mathbb{Z}$ and note that

$$\begin{array}{c}\hfill k^u{\psi }^v\left(k\right)\in \left\{\begin{array}{ccc}\mathbb{B}\left\{N^{u+v+1}\right\}\hfill & \hfill ,& u+v>-1,\hfill \\ \mathbb{B}\left\{\ln N\right\}\hfill & \hfill ,& u+v=-1,\hfill \\ \mathbb{B}\left\{N^0\right\}\hfill & \hfill ,& u+v<-1.\hfill \end{array}\right.\end{array}$$

(125)

The explicit asymptotics for some summatory functions $F\left\{k^u{\psi }^v;N,1\right\}$ are given in [23,54]. Calculate renormalization functions $R_{\infty }\left(k^u{\psi }^v;p^m\right)$ according to (60) and (61),

$$\begin{array}{ccc}\hfill R_{\infty }\left(k^u{\psi }^v;p^m\right)& =& p^{m(u+v)+1}{\displaystyle \frac{{(p+1)}^v}{{(p+1)}^v+(p-1)p^v}}, u+v\ge -1,\hfill \\ \hfill R_{\infty }\left(k^u{\psi }^v;p^m\right)& =& p^{(m-1)(u+v)}{\displaystyle \frac{{(p+1)}^v}{{(p+1)}^v-p^v+p^{-u}}}, u+v<-1.\hfill \end{array}$$

(126)

We finish this section with the summatory $F\left\{{(\varphi /\psi )}^v;N,1\right\}={\sum }_{k\le N}{\varphi }^v\left(k\right){\psi }^{-v}\left(k\right)$, $v\in \mathbb{Z}$.

The asymptotics

$${\displaystyle \frac{1}{N}}F\left\{{\displaystyle \frac{\varphi }{\psi }};N,1\right\}\simeq \prod _p\left(1-{\displaystyle \frac{2}{p(p+1)}}\right)\simeq 0.4716$$

is known due to [54]. Keeping in mind ${(\varphi /\psi )}^v\in \mathbb{B}\left\{N\right\}$, calculate the corresponding renormalization function,

$$\begin{array}{c}\hfill R_{\infty }\left({\left({\displaystyle \frac{\varphi }{\psi }}\right)}^v;p^m\right)={\displaystyle \frac{p{(p-1)}^{v-1}}{{(p-1)}^{v-1}+{(p+1)}^v}},\end{array}$$

(127)

which does not depend on m.

5.2. Renormalization of Summatory Nontotient Functions

Here, we apply the renormalization approach to summatory functions and Dirichlet series involving the divisor ${\sigma }_a\left(k\right)$, prime divisor $\beta \left(k\right)$, Piltz $d_n\left(k\right)$, Abelian group enumeration $\alpha \left(k\right)$ functions, Ramanujan sum $C_q\left(n\right)$, and some of their combinations.

5.2.1. Divisor Function ${\sigma }_a\left(k\right)$ and Prime Divisor Function $\beta \left(k\right)$

The divisor function ${\sigma }_a\left(k\right)$ is defined as the sum of the ath powers of the divisors of k. For $k=p^r$, we have

$${\sigma }_a\left(p^r\right)={\displaystyle \frac{p^{a(r+1)}-1}{p^a-1}}, a\ne 0,\hspace{1em}{\sigma }_0\left(p^r\right)=r+1.$$

• $F\left\{k^{-s}{\sigma }_a;N,1\right\}$, $a>0$, $s\ge 1+a$, $k^{-s}{\sigma }_a\in \mathbb{B}\left\{N^0\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left({\displaystyle \frac{{\sigma }_a}{k^s}};p^m\right)={\displaystyle \frac{p^{a(m+1)+s}-p^{a(m+1)}-p^s+p^a}{p^{(m+1)s}(p^a-1)}}.\end{array}$$

  (128)
• $F\left\{{\sigma }_a;N,1\right\}$, $a>0$, ${\sigma }_a\in \mathbb{B}\left\{N^{a+1}\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left({\sigma }_a;p^m\right)={\displaystyle \frac{p^{a(m+1)+1}-p^{am}-p+1}{p(p^a-1)}}.\end{array}$$

  (129)
• $F\left\{{\sigma }_a;N,1\right\}$, $a<0$, ${\sigma }_a\in \mathbb{B}\left\{N\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left({\sigma }_a;p^m\right)={\displaystyle \frac{p^{am+1}-p^{am}-p^{1-a}+1}{p^{1-a}(p^a-1)}}.\end{array}$$

  (130)
• $F\left\{{\sigma }_0^n;N,1\right\}$, ${\sigma }_0^n\in \mathbb{B}\left\{N{\left(\log N\right)}^{2^n-1}\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left({\sigma }_0^n;p^m\right)={\displaystyle \frac{\mathcal{S}(n,p,m+1)}{\mathcal{S}(n,p,1)}}, \mathcal{S}(n,p,t)=\sum _{k=1}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)t^{n-k}L{i}_{-k}\left(p^{-1}\right)+{\displaystyle \frac{t^n}{1-p^{-1}}},\end{array}$$

  (131)

  where $L{i}_{-k}\left(x\right)$ is the polylogarithm defined in (34). Substituting $n=1$ into (131), and keeping in mind $L{i}_{-1}\left(x\right)=x/{(1-x)}^2$ [48], we obtain

  $$R_{\infty }\left({\sigma }_0;p^m\right)=m+1-{\displaystyle \frac{m}{p}}.$$

  According to (129) and (130), this expression coincides with both limits of $R_{\infty }\left({\sigma }_a;p^m\right)$, when $a\to 0$, for ${\sigma }_a>$ and ${\sigma }_a<0$, respectively.
• $F\left\{{\sigma }_0{\sigma }_a;N,1\right\}$, $a>0$, ${\sigma }_0{\sigma }_a\in \mathbb{B}\left\{N^{a+1}\log N\right\}$

  $$\begin{array}{ccc}& & R_{\infty }\left({\sigma }_0{\sigma }_a;p^m\right)=\hfill \\ & & {\displaystyle \frac{p^{am+1}{\left(p^{a+1}-1\right)}^2-p^{a+1}{(p-1)}^2+m(p-1)\left(p^{a+1}-1\right)\left(p^{am}\left(p^{a+1}-1\right)-p+1\right)}{p\left(p^a-1\right)\left(p^{a+2}-1\right)}},\hfill \end{array}$$

  such that $R_{\infty }\left({\sigma }_0{\sigma }_a;1\right)=1$. Note that $R_{\infty }\left({\sigma }_0{\sigma }_a;p^m\right)\stackrel{a\to 0}{\to }{R}_{\infty }\left({\sigma }_0^2;p^m\right)$ according to (131).
• $F\left\{{\sigma }_a^2;N,1\right\}$, $a>0$, ${\sigma }_a^2\in \mathbb{B}\left\{N^{2a+1}\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left({\sigma }_a^2;p^m\right)={\displaystyle \frac{(p-1)\left(p^{a+1}-1\right)+p^{am}\left(p^{1+2a}-1\right)\left(p^{am}\left(p^{1+a}-1\right)-2(p-1)\right)}{p{\left(p^a-1\right)}^2\left(p^{a+1}+1\right)}},\end{array}$$

  (132)

  that gives according to (131)

  $$R_{\infty }\left({\sigma }_a^2;p^m\right)\stackrel{a\to 0}{\to }{R}_{\infty }\left({\sigma }_0^2;p^m\right)={\displaystyle \frac{p+{(m(p-1)+p)}^2}{p(p+1)}}.$$
• $F\left\{1/{\sigma }_0;N,1\right\}$, $1/{\sigma }_0\in \mathbb{B}\left\{N/\sqrt{\log N}\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left({\displaystyle \frac{1}{{\sigma }_0}};p^m\right)=-{\displaystyle \frac{{}_{2}F_1\left(m+1,1;m+2;p^{-1}\right)}{(m+1)p\ln \left(1-p^{-1}\right)}},\end{array}$$

  (133)

  where ${}_{2}F_1(a,b;c;z)$ denotes a generalized hypergeometric function [48]. By (133), we obtain, for $m=1$,

  $$\begin{array}{c}\hfill R_{\infty }\left({\displaystyle \frac{1}{{\sigma }_0}};p\right)=p+{\displaystyle \frac{1}{\ln \left(1-p^{-1}\right)}}, R_{\infty }\left({\displaystyle \frac{1}{{\sigma }_0}};p\right)\stackrel{p\to \infty }{⟶}{\displaystyle \frac{1}{2}}, R_{\infty }\left({\displaystyle \frac{1}{{\sigma }_0}};p\right)\stackrel{p\to 1}{⟶}1.\end{array}$$

  (134)
  Expanding an expression (133) as an infinite series ${\sum }_{r=0}^{\infty }{L}_r\left(1/{\sigma }_0;p^m\right)p^{-r}$ in accordance with Theorem 1, one can calculate $L_r\left(1/{\sigma }_0;p^m\right)$ and verify that for $0\le r\le 3$, they coincide with those given in (19).
• $F\left\{{\sigma }_1/{\sigma }_0;N,1\right\}$, ${\sigma }_1/{\sigma }_0\in \mathbb{B}\left\{N^2/\sqrt{\log N}\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left({\displaystyle \frac{{\sigma }_1}{{\sigma }_0}};p^m\right)={\displaystyle \frac{p^{m+1}{}_{2}F_1\left(m+1,1;m+2;p^{-1}\right)-{}_{2}F_1\left(m+1,1;m+2;p^{-2}\right)}{(m+1)p^2\log \left(1+p^{-1}\right)}},\end{array}$$

  (135)

  that gives for $m=1$

  $$\begin{array}{c}\hfill R_{\infty }\left({\displaystyle \frac{{\sigma }_1}{{\sigma }_0}};p\right)=p^2-{\displaystyle \frac{p-1}{\log \left(1+p^{-1}\right)}},\hspace{1em}{\displaystyle \frac{1}{p}}{R}_{\infty }\left({\displaystyle \frac{{\sigma }_1}{{\sigma }_0}};p\right)\stackrel{p\to \infty }{⟶}{\displaystyle \frac{1}{2}}.\end{array}$$

  (136)
• $F\left\{{\sigma }_1/\varphi ;N,1\right\}$, $F\left\{{\sigma }_1/\psi ;N,1\right\}$, $m\ge 1$${\sigma }_1/\varphi ,{\sigma }_1/\psi \in \mathbb{B}\left\{N\right\}$.

  $$\begin{array}{c}\hfill R_{\infty }\left({\displaystyle \frac{{\sigma }_1}{\varphi }};p^m\right)={\displaystyle \frac{p^3\left(1+p-p^{-m}\right)}{p^4-p^3+p^2+p-1}},\hspace{1em}{R}_{\infty }\left({\displaystyle \frac{{\sigma }_1}{\psi }};p^m\right)={\displaystyle \frac{p^3\left(1+p-p^{-m}\right)}{p^4+p^3-p^2-p+1}}.\end{array}$$

  (137)
  The prime divisor function $\beta \left(k\right)$ is defined by formula $\beta \left(p^{a_1}·\dots ·p^{a_n}\right)=a_1·\dots ·a_n$.
• $F\left\{\beta ;N,1\right\}$, $\beta \in \mathbb{B}\left\{N\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left(\beta ;p^m\right)=p{\displaystyle \frac{m(p-1)+1}{p^2-p+1}}.\end{array}$$

  (138)
  Let us note a curious consequence of (138) when $m=p$: $R_{\infty }\left(\beta ;p^p\right)=p$.
• $F\left\{k^{-s}\beta ;N,1\right\}$, $\beta \in \mathbb{B}\left\{N^0\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left({\displaystyle \frac{\beta }{k^s}};p^m\right)=p^{(1-m)s}{\displaystyle \frac{m(p^s-1)+1}{p^{2s}-p^s+1}}.\end{array}$$

  (139)
  A generalized summatory function $F\left\{{\beta }^n;N,1\right\}$ was considered in [35],

  $$\mathcal{F}\left({\beta }^n\right)=\prod _{p\ge 2}\left(1+\sum _{j=2}^{\infty }{\displaystyle \frac{j^n-{(j-1)}^n}{p^j}}\right),$$

  e.g.,

  $$\begin{array}{c}\hfill \mathcal{F}\left(\beta \right)=\prod _{p\ge 2}\left(1+{\displaystyle \frac{1}{p(p-1)}}\right)={\displaystyle \frac{\zeta \left(2\right)\zeta \left(3\right)}{\zeta \left(6\right)}}.\end{array}$$
• $F\left\{{\beta }^n;N,1\right\}$, ${\beta }^n\in \mathbb{B}\left\{N\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left({\beta }^n;p^m\right)={\displaystyle \frac{\mathcal{S}(n,p,m)}{1+L{i}_{-n}\left(p^{-1}\right)}},\end{array}$$

  (140)

  where $\mathcal{S}(n,p,m)$ is defined in (131).

5.2.2. Piltz Function $d_n\left(k\right)$ and the Sum of Two Squares Function $r_2\left(k\right)$

The Piltz function $d_n\left(k\right)$ is defined as a number of ways to write the positive integer k as a product of n (positive integer) factors. For $k=p^r$, we have $d_n\left(p^r\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{n+r-1}{r}}\right)$. By definition, $d_1\left(k\right)=1$ and $d_2\left(k\right)={\sigma }_0\left(k\right)$.

• $F\left\{k^{-s}{d}_n;N,1\right\}$, $d_n\in \mathbb{B}\left\{N^0\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left({\displaystyle \frac{d_n}{k^s}};p^m\right)=p^{-sm}\left({\displaystyle \genfrac{}{}{0pt}{}{n+m-1}{m}}\right){}_{2}F_1\left(m,1-n;m+1;p^{-s}\right).\end{array}$$

  (141)
• $F\left\{d_n;N,1\right\}$, $d_n\in \mathbb{B}\left\{N{\left(\log N\right)}^{n-1}\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left(d_n;p^m\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{n+m-1}{m}}\right){}_{2}F_1\left(m,1-n;m+1;p^{-1}\right).\end{array}$$

  (142)
  Note that $R_{\infty }\left(d_2;p^m\right)$ coincides with $R_{\infty }\left({\sigma }_0;p^m\right)$, given in Section 5.2.1.
• $F\left\{d_n^2;N,1\right\}$, $d_n^2\in \mathbb{B}\left\{N{\left(\log N\right)}^{n^2-1}\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left(d_n^2;p^m\right)={\left({\displaystyle \genfrac{}{}{0pt}{}{n+m-1}{m}}\right)}^2{\displaystyle \frac{{}_{3}F_2\left(1,m+n,m+n;m+1,m+1;p^{-1}\right)}{{}_{2}F_1\left(n,n;1;p^{-1}\right)}},\end{array}$$

  (143)

  and $R_{\infty }\left(d_2^2;p^m\right)=R_{\infty }\left({\sigma }_0^2;p^m\right)$ in accordance with (131).
• $F\left\{1/d_n;N,1\right\}$, $1/d_n\in \mathbb{B}\left\{N{\left(\log N\right)}^{1/n-1}\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left({\displaystyle \frac{1}{d_n}};p^m\right)=np{\left({\displaystyle \genfrac{}{}{0pt}{}{n+m-1}{m}}\right)}^{-1}{\displaystyle \frac{{}_{2}F_1\left(1,m+1;m+n;p^{-1}\right)}{np+{}_{2}F_1\left(2,1;n+1;p^{-1}\right)}},\end{array}$$

  (144)

  such that $R_{\infty }\left(d_2^{-1};p^m\right)=R_{\infty }\left({\sigma }_0^{-1};p^m\right)$ in accordance with (133).
  The number of representations of k by two squares, allowing zeros and distinguishing signs and order, is denoted by $r_2\left(k\right)$. If $k=p^r$, then

  $$\begin{array}{ccc}& & r_2\left(p^r\right)=\left\{\begin{array}{ccc}4(r+1)\hfill & \hfill ,& p=1\left(\mathrm{mod}4\right),\hfill \\ 4\hfill & \hfill ,& p=2\hfill \end{array}\right.\hfill \\ & & r_2\left(p^r\right)=\left\{\begin{array}{ccc}4\hfill & \hfill ,& p=3\left(\mathrm{mod}4\right),2\mid r\hfill \\ 0\hfill & \hfill ,& p=3\left(\mathrm{mod}4\right),2\nmid r\hfill \end{array}\right.\hfill \end{array}$$
• $F\left\{r_2;N,1\right\}$, $r_2\in \mathbb{B}\left\{N\right\}$

  $$\begin{array}{ccc}& & R_{\infty }\left(r_2;p^m\right)=\left\{\begin{array}{ccc}m+1-m/p\hfill & \hfill ,& p=1\left(\mathrm{mod}4\right),\hfill \\ 1\hfill & \hfill ,& p=2,\hfill \end{array}\right.\hfill \\ & & R_{\infty }\left(r_2;p^m\right)=\left\{\begin{array}{ccc}1\hfill & \hfill ,& p=3\left(\mathrm{mod}4\right), 2\mid m\hfill \\ 1/p\hfill & \hfill ,& p=3\left(\mathrm{mod}4\right), 2\nmid m\hfill \end{array}\right.\hfill \end{array}$$
• $F\left\{r_2/k;N,1\right\}$, $r_2/k\in \mathbb{B}\left\{\log N\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left({\displaystyle \frac{r_2}{k}};p^m\right)=p^{-m}{R}_{\infty }\left(r_2;p^m\right).\end{array}$$

  (145)

5.2.3. Ramanujan Sum $C_q\left(n\right)$ and Abelian Group Enumeration Function $\alpha \left(k\right)$

Ramanujan’s sum $C_q\left(n\right)$, $q,n\ge 1$ is a multiplicative arithmetic function, which is defined by formula $C_q\left(n\right)={\sum }_{a=1}^q\exp (2i\pi na/q)$, $(a,q)=1$ such that $C_{q_1{q}_2}\left(n\right)=C_{q_1}\left(n\right)C_{q_2}\left(n\right)$ if $(q_1,q_2)=1$ and $C_1\left(n\right)=1$. If $q=p^r$, then

$$\begin{array}{ccc}& & C_{p^r}\left(n\right)=0, if p^{r-1}\nmid n;\hspace{1em}{C}_{p^r}\left(n\right)=-p^{r-1}, if p^{r-1}\mid n, p^r\nmid n;\hfill \\ & & C_{p^r}\left(n\right)=\varphi \left(r\right), if p^r\mid n;\hspace{1em}{C}_p\left(n\right)=-1, if p\nmid n;\hfill \\ & & C_{p^r}\left(n\right)=0, if p\nmid n \mathrm{and} r\ge 2.\hfill \end{array}$$

(146)

Consider the Dirichlet summatory functions for $C_q\left(n\right)$,

$$F\left\{C_q\left(n\right)q^{-s};\infty ,1,s\right\}=\sum _{q=1}^{\infty }{C}_q\left(n\right)q^{-s}, s>1,$$

where n is kept constant. They converge to $n^{-s+1}{\sigma }_{s-1}\left(n\right){\zeta }^{-1}\left(s\right)$. We find a rescaled summatory function $F\left\{C_q\left(n\right)q^{-s};\infty ,p^m,s\right\}={\sum }_{q=1}^{\infty }{C}_{p^{m}q}\left(n\right)q^{-s}$.

Let a number n is such that $p^a\mid n$ but $p^{a+1}\nmid n$, $a\in {\mathbb{Z}}_{+}$, and $a\ge m\ge 1$, then

$$\begin{array}{ccc}& & \mathbf{U}\left({\displaystyle \frac{C_q\left(n\right)}{q^s}};p^m\right)=p^{-ms}\left(\sum _{r=m}^a{\displaystyle \frac{\varphi \left(p^r\right)}{p^{sr}}}-{\displaystyle \frac{p^a}{p^{s(a+1)}}}\right),\hfill \\ & & \mathbf{V}\left({\displaystyle \frac{C_q\left(n\right)}{q^s}};p\right)=\sum _{r=1}^a{\displaystyle \frac{\varphi \left(p^r\right)}{p^{sr}}}-{\displaystyle \frac{p^a}{p^{s(a+1)}}}.\hfill \end{array}$$

(147)

Thus, we get

$$\begin{array}{c}\hfill F\left\{{\displaystyle \frac{C_q\left(n\right)}{q^s}};\infty ,p^m,s\right\}={\displaystyle \frac{p^{-ms}}{p^s-1}}{\displaystyle \frac{W(p,s,m,a)}{p^{(1-s)(a+1)}-1}}F\left\{{\displaystyle \frac{C_q\left(n\right)}{q^s}};\infty ,1,s\right\},\\ \hfill \\ \hfill W(p,s,m,a)=(p^s-1)p^{(1-s)(a+1)}-(p-1)p^{(1-s)(m-1)}.\end{array}$$

(148)

If $a\ge m=0$, then by (147), we have $\mathbf{U}\left(C_q\left(n\right)q^{-s};1\right)=1+\mathbf{V}\left(C_q\left(n\right)q^{-s};1\right)$, and, therefore, $F\left\{C_q\left(n\right)q^{-s};\infty ,p^m,s\right\}\stackrel{m\to 0}{\to }F\left\{C_q\left(n\right)q^{-s};\infty ,1,s\right\}$. If $m\ge 2$ and the number n is not divided by p, i.e., $a=0$, then $F\left\{C_q\left(n\right)q^{-s};\infty ,p^m,s\right\}=0$.

Finally, if $m=1$ and $a=0$, then

$$F\left\{{\displaystyle \frac{C_q\left(n\right)}{q^s}};\infty ,p,s\right\}={\displaystyle \frac{1}{p^s\left(1-p^s\right)}}F\left\{{\displaystyle \frac{C_q\left(n\right)}{q^s}};\infty ,1,s\right\}.$$

Abelian group enumeration function $\alpha \left(k\right)$ accounts for the number of (isomorphism classes of) commutative groups of order k. By definition, it satisfies $\alpha \left(p^r\right)=\mathcal{P}\left(r\right)$, $\mathcal{P}\left(0\right)=1$, where $\mathcal{P}\left(r\right)$ denotes a partition function [55] with an explicit expression given in [56].

• $F\left\{\alpha ;N,1\right\}$, $\alpha \in \mathbb{B}\left\{N\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left(\alpha ;p^m\right)=p^m\left(1-Q\left(p^{-1}\right)\sum _{k=0}^{m-1}\mathcal{P}\left(k\right)p^{-k}\right), Q\left(x\right)=\prod _{j=1}^{\infty }\left(1-x^j\right), \left|x\right|\le 1.\end{array}$$

  (149)
• $F\left\{1/\alpha ;N,1\right\}$, $1/\alpha \in \mathbb{B}\left\{N\right\}$

  $$\begin{array}{c}\hfill R_{\infty }\left({\displaystyle \frac{1}{\alpha }};p^m\right)=p^m{\displaystyle \frac{\mathcal{T}(p,m)}{\mathcal{T}(p,0)}},\hspace{1em}\mathcal{T}(p,t)=\sum _{k=t}^{\infty }{\displaystyle \frac{p^{-k}}{\mathcal{P}\left(k\right)}}.\end{array}$$

  (150)

5.3. Renormalization of Summatories Associated with Ramanujan’s $\tau$ Function

The Ramanujan $\tau$ function is a multiplicative arithmetic function that is mainly known due to its generating function,

$$\sum _{k=1}^{\infty }\tau \left(k\right)x^k=x\prod _{k=1}^{\infty }{(1-x^k)}^{24}, \left|x\right|<1.$$

For our purpose of calculating the renormalization function for any summatory function $F\left\{f\right(\tau ,k);N,1\}$ with $f(\tau ,k)$, including the $\tau$ function, it is important to know its recursive relation for $k=p^r$,

$$\begin{array}{c}\hfill \tau \left(p^r\right)=\sum _{j=0}^{⌊r/2⌋}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{r-j}{r-2j}}\right)p^{11j}{\tau }^{r-2j}\left(p\right).\end{array}$$

(151)

Then, making use of Formulas (61) and (62) we can arrive at $R_{\infty }\left(f(\tau ,k);p^m\right)$ due to the finite computational procedure. However, the representation (151) is too difficult to make it worthwhile, so we choose another way to find $R_{\infty }\left(f(\tau ,k);p^m\right)$, namely by Theorem 1 and by calculating the characteristic functions $L_r\left(\tau ;p^m\right)$.

Start with identity for the $\tau$ function [9]

$$\begin{array}{c}\hfill \tau \left(p^{r+1}\right)=\tau \left(p\right)\tau \left(p^r\right)-p^{11}\tau \left(p^{r-1}\right).\end{array}$$

(152)

The following proposition is based on recursion (16) and the last identity.

Proposition 2.

$$\begin{array}{c}\hfill L_0\left(\tau ;p^m\right)=\tau \left(p^m\right), L_1\left(\tau ;p^m\right)=-p^{11}\tau \left(p^{m-1}\right), L_r\left(\tau ;p^m\right)=0, r\ge 2.\end{array}$$

(153)

Proof. 

Calculating the four first expressions of $L_r\left(\tau ;p^m\right)$, one can verify that (153) holds, i.e., $L_2\left(\tau ;p^m\right)=L_3\left(\tau ;p^m\right)=0$. We prove by induction that $L_r\left(\tau ;p^m\right)=0$, $r\ge 2$.

Indeed, let $L_q\left(\tau ;p^m\right)=0$ for $2\le q\le r$; then, keeping in mind (16) and (152), write this equality in another representation,

$$\begin{array}{c}\hfill L_q\left(\tau ;p^m\right)=\tau \left(p^{m+q}\right)-\tau \left(p^m\right)\tau \left(p^q\right)+p^{11}\tau \left(p^{m-1}\right)\tau \left(p^{q-1}\right)=0, 2\le q\le r.\end{array}$$

(154)

Making use of (152) and (154), write the next term $L_{r+1}\left(\tau ;p^m\right)$,

$$\begin{array}{c}\hfill L_{r+1}\left(\tau ;p^m\right)=\tau \left(p^{m+r+1}\right)-\tau \left(p^m\right)\tau \left(p^{r+1}\right)+p^{11}\tau \left(p^{m-1}\right)\tau \left(p^r\right),\end{array}$$

and calculate a difference,

$$\begin{array}{ccc}& & L_{r+1}\left(\tau ;p^m\right)-\tau \left(p\right)L_r\left(\tau ;p^m\right)=\tau \left(p^{m+r+1}\right)-\tau \left(p\right)\tau \left(p^{m+r}\right)-\hfill \\ & & \tau \left(p^m\right)\left[\tau \left(p^{r+1}\right)-\tau \left(p\right)\tau \left(p^r\right)\right]+p^{11}\tau \left(p^{m-1}\right)\left[\tau \left(p^r\right)-\tau \left(p\right)\tau \left(p^{r-1}\right)\right].\hfill \end{array}$$

(155)

By identity (152), the r.h.s. in equality (155) can be reduced as follows:

$$\begin{array}{ccc}& & L_{r+1}\left(\tau ;p^m\right)-\tau \left(p\right)L_r\left(\tau ;p^m\right)=\hfill \\ & & -p^{11}\left[\tau \left(p^{m+r-1}\right)-\tau \left(p^m\right)\tau \left(p^{r-1}\right)-p^{11}\tau \left(p^{m-1}\right)\tau \left(p^{r-2}\right)\right].\hfill \end{array}$$

Comparing the last equality with (154), one can recognize the function $L_{r-1}\left(\tau ;p^m\right)$ in the brackets of the last expression. Then, combining this fact with (155) and assumption (154), we obtain

$$\begin{array}{c}\hfill L_{r+1}\left(\tau ;p^m\right)=\tau \left(p\right)L_r\left(\tau ;p^m\right)-p^{11}{L}_{r-1}\left(\tau ;p^m\right)=0.\end{array}$$

(156)

Thus, the proof is finished. □

An important by-product of Proposition 2 is identity (154), which generalizes identity (152).

Corollary 2.

The Ramanujan τ function satisfies an identity,

$$\begin{array}{c}\hfill \tau \left(p^{m+n}\right)=\tau \left(p^m\right)\tau \left(p^n\right)-p^{11}\tau \left(p^{m-1}\right)\tau \left(p^{n-1}\right), m,n\ge 1.\end{array}$$

(157)

The last statement implies two inequalities that can be easily verified. The first inequality looks pretty simple, $\tau \left(p^{2n}\right)<{\tau }^2\left(p^n\right)$. Regarding the second inequality, let $p_{*}$ and $n_{*}$ be chosen in such a way that $\tau \left(p_{*}^{2n_{*}}\right)<0$, e.g., $\tau \left(2^2\right),\tau \left(3^2\right),\tau \left(5^2\right),\tau \left(7^2\right)<0$, $\tau \left(5^4\right),\tau \left({11}^4\right)<0$, etc. Then, the following inequality holds,

$$\begin{array}{c}\hfill |\tau \left(p_{*}^{2n_{*}}\right)|<p_{*}^{11}{\tau }^2\left(p_{*}^{n_{*}-1}\right).\end{array}$$

(158)

In the case $n_{*}=2$, let us combine (158) with Deligne’s inequality

$$\begin{array}{c}\hfill \left|\tau \left(p\right)\right|<2p^{11/2},\end{array}$$

(159)

for the Ramanujan $\tau$ functions [57] and obtain

$$\begin{array}{c}\hfill |\tau \left(p_{*}^4\right)|<4p_{*}^{22}.\end{array}$$

(160)

Note that the upper bound in (160) is stronger than the bound which came by combining (152) and (159) for arbitrary prime p. Indeed,

$$\begin{array}{c}\hfill \tau \left(p^4\right)={\tau }^4\left(p\right)-3p^{11}{\tau }^2\left(p\right)+p^{22}<17p^{22}-3p^{11}{\tau }^2\left(p\right).\end{array}$$

(161)

However, according to (159),

$$17p^{22}-3p^{11}{\tau }^2\left(p\right)>17p^{22}-12p^{22}>4p^{22}.$$

Numerical calculations show that inequality (160) is also valid for the first 474 primes, regardless of whether the requirement $\tau \left(p^4\right)<0$ is met or not. Inequality (160) is violated for the first time for the 475th prime,

$$\begin{array}{c}\hfill p_{475}=3371, {\displaystyle \frac{\tau \left({3371}^4\right)}{4·{3371}^{22}}}\simeq 1.0119.\end{array}$$

5.3.1. Renormalization of the Ramanujan $\tau$ Dirichlet Series

Recalling the Ramanujan conjecture on the $\tau$ function, $\tau \left(N\right)=\mathcal{O}\left(N^{11/2+\epsilon }\right)$, proved by Deligne [57], consider the $\tau$ Dirichlet series $F\left\{\tau ·k^{-s};\infty ,p^m\right\}={\sum }_{k=1}^{\infty }\tau \left(k\right)k^{-s}$, where $\tau ·k^{-s}\in \mathbb{B}\left\{N^0\right\}$. The series converges absolutely if $s>13/2$. By Theorem 1 and relationship (112) between characteristic functions for $\tau \left(k\right)$ and $\tau \left(k\right)k^{-s}$, the renormalization function $R_{\infty }\left(\tau ·k^{-s};p^m\right)$ reads

$$\begin{array}{c}\hfill R_{\infty }\left(\tau ·k^{-s},p^m\right)=\sum _{r=0}^{\infty }{L}_r\left(\tau ·k^{-s};p^m\right)=\sum _{r=0}^{\infty }{L}_r\left(\tau ;p^m\right)p^{-s(r+m)}.\end{array}$$

(162)

Applying Proposition 2 to (162), we obtain

$$\begin{array}{c}\hfill R_{\infty }\left(\tau ·k^{-s},p^m\right)=p^{-sm}\left(\tau \left(p^m\right)-{\displaystyle \frac{\tau \left(p^{m-1}\right)}{p^{s-11}}}\right).\end{array}$$

Making use of the relation between the renormalization function $R_{\infty }\left(f·k^{-s};p^m\right)$ and a ratio $\mathfrak{D}\left(f;p^m,s\right)$ between scaled and unscaled Dirichlet series, given in Section 4.2, we finally obtain

$$\begin{array}{c}\hfill \sum _{k=1}^{\infty }{\displaystyle \frac{\tau \left(k{p}^m\right)}{k^s}}=\left(\tau \left(p^m\right)-{\displaystyle \frac{\tau \left(p^{m-1}\right)}{p^{s-11}}}\right)\sum _{k=1}^{\infty }{\displaystyle \frac{\tau \left(k\right)}{k^s}}.\end{array}$$

(163)

Formula (163) gives rise to several special cases, e.g.,

$$\begin{array}{c}\hfill \sum _{k=1}^{\infty }{\displaystyle \frac{\tau \left(kp\right)}{k^s}}=\left(\tau \left(p\right)-p^{11-s}\right)\sum _{k=1}^{\infty }{\displaystyle \frac{\tau \left(k\right)}{k^s}},\sum _{n=0}^m\sum _{k=1}^{\infty }{\displaystyle \frac{\tau \left(k{p}^n\right)}{k^{11}}}=\tau \left(p^m\right)\sum _{k=1}^{\infty }{\displaystyle \frac{\tau \left(k\right)}{k^{11}}}.\end{array}$$

5.3.2. Renormalization Functions $R_{\infty }\left({\tau }^2,p^m\right)$ and $R_{\infty }\left({\tau }^2·k^{-25/2},p^m\right)$

In this section, we also renormalize two summatory functions, $F\left\{{\tau }^2;N,p^m\right\}$ and $F\left\{{\tau }^2·k^{-25/2};N,p^m\right\}$, presented in Table 1 and Table 2. For this purpose, we proceed to calculate the characteristic functions $L_r\left({\tau }^2;p^m\right)$. Unlike $L_r\left(\tau ;p^m\right)$ described in Proposition 2, this case is not so simple, but it still allows finding general expressions.

Proposition 3.

$$\begin{array}{ccc}& & L_0\left({\tau }^2;p^m\right)={\tau }^2\left(p^m\right),\hfill \\ & & L_1\left({\tau }^2;p^m\right)=p^{22}{\tau }^2\left(p^{m-1}\right)-2p^{11}\tau \left(p\right)\tau \left(p^{m-1}\right)\tau \left(p^m\right),\hfill \\ & & L_r\left({\tau }^2;p^m\right)=2{\left(-p^{11}\right)}^r\tau \left(p\right)\tau \left(p^{m-1}\right)\tau \left(p^m\right), r\ge 2.\hfill \end{array}$$

(164)

Proof. 

Let us prove the proposition by induction. First, we calculate the first five expressions of $L_r\left({\tau }^2;p^m\right)$ and verify that (164) holds. Let the proposition be true for $2\le q\le r$, then prove that

$$\begin{array}{c}\hfill L_{r+1}\left({\tau }^2;p^m\right)=2{\left(-p^{11}\right)}^{r+1}\tau \left(p\right)\tau \left(p^{m-1}\right)\tau \left(p^m\right).\end{array}$$

(165)

Keeping in mind (16), calculate $L_{r+1}\left({\tau }^2;p^m\right)$,

$$\begin{array}{ccc}& & L_{r+1}\left({\tau }^2;p^m\right)=\hfill \\ & & {\tau }^2\left(p^{m+r+1}\right)-{\tau }^2\left(p^{r+1}\right){\tau }^2\left(p^m\right)-{\tau }^2\left(p^r\right)p^{11}\tau \left(p^{m-1}\right)\left[p^{11}\tau \left(p^{m-1}\right)-2\tau \left(p\right)\tau \left(p^m\right)\right]-\hfill \\ & & 2\tau \left(p\right)\tau \left(p^{m-1}\right)\tau \left(p^m\right)p^{22}\left[{\tau }^2\left(p^{r-1}\right)-{\tau }^2\left(p^{r-2}\right)p^{11}+\dots +{(-1)}^{r+1}{\tau }^2\left(p\right)p^{11(r-2)}\right].\hfill \end{array}$$

By (152), the first four terms in the above equality are reduced to a single term,

$$\begin{array}{c}\hfill {\tau }^2\left(p^{m+r+1}\right)-{\tau }^2\left(p^{r+1}\right){\tau }^2\left(p^m\right)-{\tau }^2\left(p^r\right)p^{11}\tau \left(p^{m-1}\right)\left[p^{11}\tau \left(p^{m-1}\right)-2\tau \left(p\right)\tau \left(p^m\right)\right]=\\ \hfill 2\tau \left(p^m\right)\tau \left(p^{m-1}\right)\tau \left(p^r\right)p^{11}\left[\tau \left(p^r\right)\tau \left(p\right)-\tau \left(p^{r+1}\right)\right]=2\tau \left(p^m\right)\tau \left(p^{m-1}\right)\tau \left(p^r\right)\tau \left(p^{r-1}\right)p^{22},\end{array}$$

which simplifies further calculations,

$$\begin{array}{c}\hfill L_{r+1}\left({\tau }^2;p^m\right)=2\tau \left(p^m\right)\tau \left(p^{m-1}\right)p^{22}{A}_{r-1},\end{array}$$

(166)

where

$$\begin{array}{c}\hfill A_{r-1}=\tau \left(p^{r-1}\right)\left[\tau \left(p^r\right)-\tau \left(p\right)\tau \left(p^{r-1}\right)\right]+\tau \left(p\right){\tau }^2\left(p^{r-2}\right)p^{11}-\dots \pm {\tau }^3\left(p\right)p^{11(r-2)}.\end{array}$$

Performing calculations in the brackets by (152), we obtain

$$\begin{array}{c}\hfill L_{r+1}\left({\tau }^2;p^m\right)=-2\tau \left(p^m\right)\tau \left(p^{m-1}\right)p^{33}{A}_{r-2},\end{array}$$

(167)

where

$$\begin{array}{c}\hfill A_{r-2}=\tau \left(p^{r-2}\right)\left(\tau \left(p^{r-1}\right)-\tau \left(p\right)\tau \left(p^{r-2}\right)\right)+\tau \left(p\right){\tau }^2\left(p^{r-3}\right)p^{11}-\dots \pm {\tau }^3\left(p\right)p^{11(r-3)}.\end{array}$$

Comparing (166) and (167) and continuing to contract the terms in brackets, we obtain

$$\begin{array}{ccc}\hfill L_{r+1}\left({\tau }^2;p^m\right)& =& {(-1)}^{r}2\tau \left(p^m\right)\tau \left(p^{m-1}\right)p^{11r}{A}_1,\hfill \\ \hfill A_1& =& \tau \left(p\right)\left(\tau \left(p^2\right)-{\tau }^2\left(p\right)\right)=-\tau \left(p\right)p^{11}.\hfill \end{array}$$

Combining both equalities, we arrive at (165), and the proposition is proven. □

Calculate the renormalization function $R_{\infty }\left({\tau }^2,p^m\right)$. According to Table 1, we obtain ${\tau }^2\in \mathbb{B}\left\{N^{12}\right\}$, i.e., $a_1=12$. Then, by Theorem 1 and Proposition 3, we obtain

$$\begin{array}{c}\hfill R_{\infty }\left({\tau }^2,p^m\right)={\tau }^2\left(p^m\right)+p^{10}{\tau }^2\left(p^{m-1}\right)-{\displaystyle \frac{2\tau \left(p\right)}{p+1}}\tau \left(p^{m-1}\right)\tau \left(p^m\right),\end{array}$$

(168)

such that $R_{\infty }\left({\tau }^2,1\right)=1$. In the case $m=1$, we obtain

$$R_{\infty }\left({\tau }^2,p\right)=p^{10}+{\displaystyle \frac{p-1}{p+1}}{\tau }^2\left(p\right).$$

Finally, calculate the renormalization function $R_{\infty }\left({\tau }^2·k^{-25/2},p^m\right)$, keeping in mind that, according to Table 1, we have ${\tau }^2·k^{-25/2}\in \mathbb{B}\left\{N^0\right\}$. Using the relationship (112) between characteristic functions $L_r\left(f·k^{-s};p^m\right)$ and $L_r\left(f;p^m\right)$, build the renormalization function as it was performed in formula (162) for the Ramanujan $\tau$ Dirichlet series,

$$\begin{array}{c}\hfill R_{\infty }\left({\displaystyle \frac{{\tau }^2}{k^{25/2}}},p^m\right)=\sum _{r=0}^{\infty }{L}_r\left({\displaystyle \frac{{\tau }^2}{k^{25/2}}};p^m\right)=p^{-25m/2}\sum _{r=0}^{\infty }{\displaystyle \frac{L_r\left({\tau }^2;p^m\right)}{p^{25r/2}}}.\end{array}$$

By Proposition 3 and the above formula, we obtain

$$\begin{array}{ccc}& & R_{\infty }\left({\displaystyle \frac{{\tau }^2}{k^{25/2}}},p^m\right)=\hfill \\ & & p^{-25m/2}\left({\tau }^2\left(p^m\right)+p^{19/2}{\tau }^2\left(p^{m-1}\right)-{\displaystyle \frac{2\tau \left(p\right)}{p^{3/2}+1}}\tau \left(p^{m-1}\right)\tau \left(p^m\right)\right).\hfill \end{array}$$

(169)

6. Numerical Verification

In this section, we verify the renormalization approach for the summation of multiplicative arithmetic functions with a scaled summation variable. Consider a relative deviation $\rho \left(f;N,p^m\right)$ between rescaled $F\left\{f;N,p^m\right\}$ and nonscaled $F\left\{f;N,1\right\}$ summatory functions,

$$\begin{array}{c}\hfill \rho \left(f;N,p^m\right)={\displaystyle \frac{1}{R_{\infty }\left(f;p^m\right)}}\left({\displaystyle \frac{F\left\{f;N,p^m\right\}}{F\{f;N,1\}}}-R_{\infty }\left(f;p^m\right)\right),\end{array}$$

(170)

where the renormalization function $R_{\infty }\left(f;p^m\right)$ is calculated according to Theorem 1. In Figure 1, Figure 2 and Figure 3, we present plots of $\rho \left(f;N,p^m\right)$ for six different arithmetic functions. These figures show that the formulas for renormalization functions work with high precision.

Figure 1. Plots of $\rho (\varphi ;N,11)$ (a) and $\rho ({\sigma }_1;N,11)$ (b) in the range $2\times {10}^4\le N\le {10}^6$.

Figure 2. Plots of $\rho ({\sigma }_{-1};N,11)$ (a) and $\rho \left(\right|\mu |;N,11)$ (b) in the range $2\times {10}^4\le N\le {10}^6$.

Figure 3. Plots of $\rho (\tau ·k^{-7};N,3)$ in the range $5\times {10}^3\le N\le {10}^5$ (a) and $\rho ({\tau }^2;N,3)$ (b) in the range $5\times {10}^3\le N\le 3·{10}^5$.

7. Concluding Remarks

The importance of asymptotic summation of divergent series of arithmetic and non-arithmetic functions $f\left(k\right)$ with a scaled summation variable was recognized in statistical physics in the 1960s–1970s. The calculation of lattice partition functions determines critical phenomena in such models, e.g., the Ising, Potts, and XXY models. Just then, 50 years ago, a summation problem was resolved by elaborating [16,17] an approach that connects the sums of two adjacent scales by a renormalization equation.

In the present paper, we apply the renormalization ideas of statistical physics to calculate asymptotically summatory multiplicative arithmetic functions $F\left\{f;N,p^m\right\}$ with a scaled summation variable, that, to the best of our knowledge, have not been discussed in the literature. The main assertion of this paper is given in Formula (5), where the impact of scaling with a prime power $p^m$ of a summation variable k in $f\left(k{p}^m\right)$ is reduced to renormalization of the summatory function $F\left\{f;N,1\right\}$ without scaling. For the ratio $F\left\{f;N,p^m\right\}/F\left\{f;N,1\right\}$, we call the renormalization function $R\left\{f;N,p^m\right\}$ and study its asymptotics $R_{\infty }\left(f,p^m\right)$ when $N\to \infty$.

We study the asymptotic renormalization function $R_{\infty }\left(f,p^m\right)$ in various aspects: (a) its existence as a convergent numerical series, (b) its multiplicativity property without specifying the function $f\left(k\right)$, (c) formulas for $R_{\infty }\left(f{k}^{-s},p^m\right)$ for corresponding Dirichlet series, (d) formulas for $R_{\infty }\left(f\left(k^n\right),p^m\right)$ for different arithmetic functions $f\left(k\right)$, (e) the multiplicative property of the renormalization functions $R_{\infty }\left(f;{\prod }_{i=1}^n{p}_i^{m_i}\right)$ with complex scaling ${\prod }_{i=1}^n{p}_i^{m_i}$ of a summation variable (see Corollary 1, Section 3.2).

We calculate $R_{\infty }\left(f,p^m\right)$ for basic multiplicative arithmetic functions, such as Euler $\varphi \left(k\right)$, Dedekind $\psi \left(k\right)$, Jordan $J_n\left(k\right)$, Liouville $\lambda \left(k\right)$, Piltz $d_n\left(k\right)$, divisor ${\sigma }_n\left(k\right)$, prime divisor $\beta \left(k\right)$, Ramanujan $\tau \left(k\right)$, and many others. Many non-multiplicative arithmetic functions, such as the prime counting function $\pi \left(k\right)$, Chebyshev $\Theta \left(k\right)$, Mangoldt $\Lambda \left(k\right)$, and the partition function $P\left(k\right)$, were left beyond the scope of this article. The study of their summation functions is a challenging task and may require further research.

In this regard, another issue worth mentioning is the possibility of applying asymptotic summation of divergent series by renormalization in models with a multiscale structure. Such structures can be found in various fields of science, such as fluid mechanics (an evolution of turbulence cascade), porous media and composites (microscopic scales at the level of pores and grains), computer science (machine learning, deep learning, and various fields of data analysis), and astrophysics (a scale-setting procedure for general relativity with renormalization group corrections).