On the Product of Zeta-Functions

Abstract

In this paper, we study the Bochner modular relation (Lambert series) for the kth power of the product of two Riemann zeta-functions with difference $\alpha$, an integer with the Voronoĭ function weight $V_k$. In the case of $V_1\left(x\right)=e^{-x}$, the results reduce to Bochner modular relations, which include the Ramanujan formula, Wigert–Bellman approximate functional equation, and the Ewald expansion. The results abridge analytic number theory and the theory of modular forms in terms of the sum-of-divisor function. We pursue the problem of (approximate) automorphy of the associated Lambert series. The $\alpha =0$ case is the divisor function, while the $\alpha =1$ case would lead to a proof of automorphy of the Dedekind eta-function à la Ramanujan.

1. Introduction

There has been considerable interest in the literature about Abel means of divergent series ${\sum }_{n=1}^{\infty }{a}_n$, e.g., (pp. 71–73, [1]). For an increasing sequence $\left\{{\lambda }_n\right\}$, ${\lambda }_1>0$, if $f\left(x\right)={\sum }_{n=1}^{\infty }{a}_n{e}^{-{\lambda }_{n}x}$ is convergent for $x>0$ and ${lim}_{x\to 0}f\left(x\right)=l$, then we say that the Abel mean of ${\sum }_{n=1}^{\infty }{a}_n$ is l or that it is Abel summable to l. Reference [2] contains many examples that have Abel means. Abel’s main contribution, however, is that the zeta-function $f_3\left(z\right)={\sum }_{n=1}^{\infty }{d}_3\left(n\right)z^n$ does not have, while Estermann’s $f_2\left(z\right)={\sum }_{n=1}^{\infty }{d}_2\left(n\right)z^n$ has, where $d_k\left(n\right)$ indicates the divisor function. We shall mainly treat the case with exponential weight and its generalization. The method of the present paper can be adopted to study similar problems for various divisor problems too. We are restricted, however, to the case of the sum-of-divisors function so as to show the connection with modular forms (when we shift to the divisor function, such a relation will be lost and will belong to another area, to which we shall return elsewhere.

Instead of an arbitrary sequence, we may restrict ourselves to a sequence whose generating Dirichlet series $\phi \left(s\right):={\sum }_{n=1}^{\infty }{a}_n{n}^{-s}$ is absolutely convergent in some half-plane and meromorphically continued over the whole plane and satisfies the functional equation. Then, interpreting the convergence factor $e^{-{\lambda }_{n}x}$ as the Mellin inverse of the gamma function, we may study the behavior of the Lambert series $f\left(x\right)$ employing the associated zeta-function $\phi \left(s\right)$. In particular, we may often express the resulting integral as another Lambert series of the form $f\left(\frac{1}{x}\right)$ (the counterpart in the Bochner modular relation), and we may view the Abel mean as a variant of the Bochner modular relation [3].

In this context, Bellman [4] studied the approximate functional equation as a generalization of [5], cf. also (Chapter 12, [6]). Independently of these, Atkinson [7] and later Berndt [8] studied Abel means, which culminated in (§5.2, pp. 154–156, [9]). Ref. [10] conducted an extensive study on the Wigert–Bellman approximate functional equation and derived the expansion into confluent hypergeometric function series, which was developed as an Ewald expansion in (Chapter 5, pp. 144–168, [9]).

In this paper, we consider Lambert series with the Voronoĭ function $V_k$ and deduce variants of the Bochner modular relation.

Recall the Riemann zeta-function defined by

$$\zeta \left(s\right)=\prod _p{\left(1-\frac{1}{p^s}\right)}^{-1}=\sum _{n=1}^{\infty }\frac{1}{n^s},\sigma :=\mathrm{Re}s>1.$$

It continues meromorphically over the whole plane with a simple pole at $s=1$ through the functional equation.

$$\zeta \left(s\right)={\pi }^{s-\frac{1}{2}}\frac{\Gamma \left(\frac{1-s}{2}\right)}{\Gamma \left(\frac{s}{2}\right)}\zeta (1-s).$$

In this paper, we consider the product of two Riemann zeta-functions

$$\phi (s,\alpha )=\zeta \left(s\right)\zeta (s+\alpha )=\sum _{n=1}^{\infty }\frac{{\sigma }_{-\alpha }\left(n\right)}{n^s},$$

(1)

and its kth power (3), where the series is absolutely convergent for $\sigma >{\sigma }_{\phi }:=max\{1,1-\mathrm{Re}\alpha \}$ and

$${\sigma }_{-\alpha }\left(n\right)=\sum _{d|n}{d}^{-\alpha }$$

(2)

is the sum-of-divisors function. We note that $\phi (s,-a)$ includes the case of $\zeta \left(s\right)\zeta (s-\alpha )$ as $\phi (s,-\alpha )={\sum }_{n=1}^{\infty }\frac{{\sigma }_{\alpha }\left(n\right)}{n^s}$. Since the results can be written down either in $\alpha$ or $-\alpha$, we stick to (1). The partial theory of modular relations for the product of zeta functions is given in (Chapter 9, pp. 241–265, [9]). Here, we state variants of the Bochner modular relation, and as a second goal, we clarify to what extent automorphy is preserved.

$\alpha$ can be any complex number, but for intended application purposes, we assume $\alpha$ is a fixed integer. This will already give quite a lot of interesting elucidating results, entailing a generalization of [4].

For special functions used in this paper, we refer to [11,12,13], etc.

We state the results for the Zeta function

$$\phi \left(s\right):=\phi {(s,\alpha )}^k={\left(\zeta \right(s\left)\zeta \right(s+\alpha \left)\right)}^k=\sum _{n=1}^{\infty }\frac{{\sigma }_{-\alpha }^k\left(n\right)}{n^s},$$

(3)

say, where $k\ge 1$ is an integer, and the series is absolutely convergent for $\sigma >{\sigma }_{\phi }$ with ${\sigma }_{\phi }$ given above in the first instance. It is continued meromorphically over the whole plane by Lemma 4 below. Here,

$${\sigma }_0^k\left(n\right)=d_{2k}\left(n\right)=\sum _{d_1···d_{2k}=n}1$$

(4)

is the $2k$-dimensional divisor function. ${\sigma }_{-\alpha }^k\left(n\right)={\sum }_{d_1···d_{2k}=n}{\sigma }_{-\alpha }\left(d_1\right)···{\sigma }_{-\alpha }\left(d_1\right)$ is the convolution of (2), and

$${\sigma }_{\alpha }^1\left(n\right)={\sigma }_{\alpha }\left(n\right).$$

For $k=1$, we derive the Ewald expansion, which explains the situation surrounding the Chapman–Koyama-Kurokawa Eisenstein series and Ramanujan’s formula.

Theorem 1.

Let $V\left(x\right)=V_k\left(x\right)$ be the Voronoĭ function (53) and let $\mathrm{P}\left(x\right)={\mathrm{P}}_{\alpha }\left(x\right)$ be the sum of residues of the integrand $\Gamma {\left(s\right)}^k\phi \left(s\right)x^{-s}$ at $-\alpha ,1-\alpha ,···,-1,0,1$ for $\alpha >0$ and at $0,1,1-\alpha$ for $\alpha \le 0$. Then,

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k(n)V(nx)\hfill \\ \hfill & =\frac{{\left({\left(2\pi \right)}^{-\alpha }{(-1)}^{\frac{\alpha }{2}}\right)}^k}{x^{1-\alpha }}\sum _{j=0}^{2k}\left(\genfrac{}{}{0pt}{}{2k}{j}\right)e^{\pi i(k-j)\alpha }\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)\hfill \\ \hfill & \times G_{k,2k}^{2k,k}\left(\frac{{\left(4{\pi }^2\right)}^k{e}^{-\pi i(k-j)}}{x}n\left|\begin{array}{c}\stackrel{k}{\overbrace{0,···,0}}\hfill \\ \underset{2k}{\underbrace{0,···,0}}\hfill \end{array}\right.\right)+\mathrm{P}\left(x\right)\hfill \end{array}$$

(5)

and

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)V\left(nx\right)\hfill \\ \hfill & ={(-1)}^{\frac{\alpha +1}{2}k}{\left(\frac{x}{{\left(2\pi \right)}^k}\right)}^{\alpha -1}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)V\left(\frac{{\left(4{\pi }^2\right)}^k}{x}n\right)+\mathrm{P}\left(x\right)+O\left(1\right),x=O\left(1\right)\hfill \end{array}$$

(6)

if α is even, while when α is odd, we have a generalization of the Bochner modular relation entailing Ramanujan’s formula

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)V\left(nx\right)\hfill \\ \hfill & ={(-1)}^{\frac{\alpha -1}{2}k}{\left(\frac{x}{{\left(2\pi \right)}^k}\right)}^{\alpha -1}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)V\left(\frac{{\left(4{\pi }^2\right)}^k}{x}n\right)+\mathrm{P}\left(x\right).\hfill \end{array}$$

(7)

The concrete form of $\mathrm{P}\left(x\right)$ (for $k=1$) is given by Lemma 1. All of these are valid for $\mathrm{Re}x>0$ and may be expressed in a more symmetric form by changing x to ${\left(2\pi \right)}^{k}x$. Asymptotic expansions are available, for example, in view of (p. 278, Theorem 1, [14]).

In the case $k=1$ and $\alpha =2n+1$, an odd integer of Theorem 1 is rather famous, Ramanujan’s formula, and we state and illustrate the basic ideas.

Let $\alpha >0,\beta >0$ satisfy the relation $\alpha \beta ={\pi }^2$ and let n be any positive integer. Then ((2.4), (2.5), [15]),

$$\begin{array}{cc}\hfill & {\alpha }^{-n}\left\{\frac{1}{2}\zeta (2n+1)+\sum _{k=1}^{\infty }\frac{k^{-2n-1}}{e^{2\alpha k}-1}\right\}\hfill \\ \hfill & ={(-\beta )}^{-n}\left\{\frac{1}{2}\zeta (2n+1)+\sum _{k=1}^{\infty }\frac{k^{-2n-1}}{e^{2\beta k}-1}\right\}\hfill \\ \hfill & -2^{2n}\sum _{j=0}^{n+1}{(-1)}^j\frac{B_{2j}}{\left(2j\right)!}\frac{B_{2n+2-2j}}{(2n+2-2j)!}{\alpha }^{n+1-j}{\beta }^j,\hfill \end{array}$$

(8)

where $B_m$ denotes the mth Bernoulli number, cf., e.g., (p. 59, [13]). Formula (8) is true for all integers n by modifying the residual function, Section 2. It is customary to put $\alpha =\pi x$ and $\beta =\pi x^{-1}$, where $\mathrm{Re}x>0$ and express (8) as a relation between the correspondence $x\leftrightarrow \frac{1}{x}$ in the right half-plane. The transition between the Lambert series and the rapidly convergent series is carried out using Liouville’s formula:

$$\sum _{m=1}^{\infty }\frac{a_m}{e^{mz}-1}=\sum _{n=1}^{\infty }{b}_n{e}^{-nz}$$

(9)

where $b_n={\sum }_{d\mid n}{a}_d$, with d running through all positive divisors of n, and the middle expression is called the Lambert series. For an integer $\alpha$, let ${\mathrm{P}}_{\alpha }\left(x\right)$ be the residual function (12). Then, we have the Bochner modular relation

$$\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-nx}={(-1)}^{\frac{\alpha -1}{2}}{\left(\frac{x}{2\pi }\right)}^{\alpha -1}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-\frac{4{\pi }^{2}n}{x}}+{\mathrm{P}}_{\alpha }\left(x\right).$$

(10)

Its upper half-plane version reads as follows, with $\tau \in \mathcal{H}$ and any integer $\varkappa \ge 0$:

$$\begin{array}{cc}\hfill & \sum _{k=1}^{\infty }{\sigma }_{-2\varkappa -1}\left(k\right)e^{2\pi ik\tau }-{\tau }^{2\varkappa }\sum _{k=1}^{\infty }{\sigma }_{-2\varkappa -1}\left(k\right)e^{-\frac{2\pi ik}{\tau }}={\mathrm{P}}_{2\varkappa +1}(-2\pi \tau ),\hfill \\ \hfill & {\mathrm{P}}_{2\varkappa +1}(-2\pi \tau )=\frac{{(-1)}^{\varkappa +1}{\left(2\pi \right)}^{2\varkappa +1}}{2i\tau }\sum _{j=0}^{\varkappa +1}{(-1)}^j\frac{B_{2j}}{\left(2j\right)!}\frac{B_{2\varkappa +2-2j}}{(2\varkappa +2-2j)!}{\tau }^{2\varkappa +2-2j}\hfill \\ \hfill & +\left\{\begin{array}{cc}-\frac{1}{2}\zeta (2\varkappa +1)\left\{1-{\tau }^{2\varkappa }\right\}\hfill & \mathrm{if}\varkappa \ge 1,\hfill \\ \frac{1}{2}log\frac{\tau }{i}\hfill & \mathrm{if}\varkappa =0.\hfill \end{array}\right.\hfill \end{array}$$

(11)

In this form, it is an example of the Hecke correspondence between the q-series and the Hecke functional Equation (52) as developed by Hecke in the 1930s–1940s. The case of the negative integer $\varkappa$ leads to the automorphy of the Eisenstein series, cf. Section 3.

2. The Product of Two Riemann Zeta-Functions

In this section, we state the case $k=1$ of Theorem 1, cf. [16].

Lemma 1.

Let

$${\mathrm{P}}_{\alpha }\left(x\right)=\sum _{s\in S}\mathrm{Res}\Gamma \left(s\right)\zeta \left(s\right)\zeta (s+\alpha )x^{-s},$$

(12)

where $S=\{-\alpha ,1-\alpha ,···,-1,0,1\}$ for $\alpha >0$ and $S=\{0,1,1-\alpha \}$ for $\alpha \le 0$. Then, for $\alpha \ge 0$,

$$\begin{array}{cc}\hfill {\mathrm{P}}_{\alpha }\left(x\right)& =\sum _{\begin{array}{c}r=1\\ 2\nmid r\ne \alpha -1\end{array}}^{\alpha }\frac{{(-1)}^r}{r!}\zeta (-r)\zeta (-r+\alpha )x^r-\frac{1}{2}(1-{\delta }_{\alpha 1})\zeta \left(\alpha \right)\hfill \\ \hfill & +\left\{\begin{array}{cc}\zeta (1+\alpha )x^{-1}+\frac{{(-1)}^{\alpha -1}}{(\alpha -1)!}{x}^{\alpha -1}({\zeta }^{\prime }(1-\alpha )-\zeta (1-\alpha )logx),\hfill & \alpha >0\hfill \\ \frac{1}{x}(\gamma -logx),\hfill & \alpha =0\hfill \end{array}\right.,\hfill \end{array}$$

(13)

where ${\delta }_{\alpha 1}$ is the Kronecker symbol and the empty sum is to be interpreted as 0. In particular,

$${\mathrm{P}}_0\left(x\right)=\frac{1}{4}+\frac{1}{x}(\gamma -logx).$$

(14)

For $\alpha <0$,

$${\mathrm{P}}_{\alpha }\left(x\right)=\Gamma (1-\alpha )\zeta (1-\alpha )x^{\alpha -1}+\zeta (1+\alpha )x^{-1}-\frac{1}{2}\zeta \left(\alpha \right).$$

(15)

In the literature, (13) is expressed in another form based on the explicit formula for zeta-values, cf., e.g., (p. 98, (18), [13]).

$$\zeta (1-2k)=-\frac{B_{2k}}{2k}(k\ge 1),\zeta \left(2k\right)={(-1)}^{k+1}\frac{2^{2k}}{2\left(2k\right)!}{B}_{2k}{\pi }^{2k}.$$

(16)

For $\alpha =2\varkappa +1>0$, (13) leads to

$$\begin{array}{cc}\hfill {\mathrm{P}}_{2\varkappa +1}\left(2\pi x\right)=& \frac{{\left(2\pi \right)}^{2\varkappa +1}}{2x}\sum _{j=0}^{\varkappa +1}{(-1)}^{j+1}\frac{B_{2j}}{\left(2j\right)!}\frac{B_{2\varkappa +2-2j}}{(2\varkappa +2-2j)!}{x}^{2\varkappa +2-2j}\hfill \\ \hfill & +\left\{\begin{array}{cc}-\frac{1}{2}\zeta (2\varkappa +1)\left\{1+{(-1)}^{\varkappa +1}{x}^{2\varkappa }\right\}\hfill & \mathrm{if}\varkappa \ge 1,\hfill \\ \frac{1}{2}logx\hfill & \mathrm{if}\varkappa =0,\hfill \end{array}\right.\hfill \end{array}$$

(17)

which corrects (p. 82, (3.88), [9]). Since we have ($1\ne \alpha =2\varkappa +1$ odd),

$$\begin{array}{cc}\hfill & {\mathrm{P}}_{\alpha }\left(x\right)=-\sum _{\begin{array}{c}r=1\\ 2\nmid r\end{array}}^{\alpha }\frac{1}{r!}\zeta (-r)\zeta (-r+\alpha )x^r+\zeta (1+\alpha )x^{-1}-\frac{1}{2}\zeta \left(\alpha \right)+\frac{{(-1)}^{\alpha -1}{\zeta }^{\prime }(1-\alpha )}{(\alpha -1)!}{x}^{\alpha -1}\hfill \\ \hfill & =-\sum _{j=0}^{\varkappa }\frac{1}{(\alpha -2j)!}\zeta (-\alpha +2j)\zeta \left(2j\right)x^{2\varkappa -2j+1}+\zeta (1+\alpha )x^{-1}-\frac{1}{2}\zeta \left(\alpha \right)\hfill \\ \hfill & +\frac{{\zeta }^{\prime }(1-\alpha )}{(\alpha -1)!}{x}^{\alpha -1}\hfill \\ \hfill & =\sum _{j=0}^{\varkappa +1}\frac{1}{(\alpha -2j+1)!}{B}_{\alpha -2j+2}\zeta \left(2j\right)x^{2\varkappa -2j+1}-\frac{1}{2}\zeta \left(\alpha \right)+\frac{{\zeta }^{\prime }(1-\alpha )}{(\alpha -1)!}{x}^{\alpha -1}\hfill \end{array}$$

(18)

on writing $r=\alpha -2j$ and noting $\zeta (1-\alpha )=0$. Using (16) and the formula (p. 99, (22), [13])

$${\zeta }^{\prime }(-2n)={(-1)}^n\frac{\left(2n\right)!}{2{\left(2\pi \right)}^{2n}}\zeta (2n+1)$$

(19)

for $n\ge 1$, we deduce that

$$\begin{array}{cc}\hfill {\mathrm{P}}_{2\varkappa +1}\left(x\right)=& \frac{{\left(2\pi \right)}^{2\varkappa +2}}{2x}\sum _{j=0}^{\varkappa +1}{(-1)}^{j+1}\frac{B_{2j}}{\left(2j\right)!}\frac{B_{2\varkappa +2-2j}}{(2\varkappa +2-2j)!}{\left(\frac{x}{2\pi }\right)}^{2\varkappa +2-2j}\hfill \\ \hfill & -\frac{1}{2}\zeta (2\varkappa +1)\left(1+{(-1)}^{\varkappa +1}{\left(\frac{x}{2\pi }\right)}^{2\varkappa }\right),\hfill \end{array}$$

(20)

which is (17). For $\varkappa =0$, (17) reads

$${\mathrm{P}}_1\left(x\right)=-\frac{\pi }{2}{B}_{2}x+\frac{\pi }{2}{B}_2{x}^{-1}+\frac{1}{2}logx.$$

(21)

Similarly, (13) with $\alpha =2\varkappa >0$ leads to

$$\begin{array}{cc}\hfill & {\mathrm{P}}_{2\varkappa }\left(x\right)=-\sum _{\begin{array}{c}r=1\\ 2\nmid r\end{array}}^{2\varkappa -1}\frac{1}{r!}\zeta (-r)\zeta (2\varkappa -r)x^r+\zeta (2\varkappa +1)x^{-1}-\frac{1}{2}\zeta \left(2\varkappa \right)-\frac{{\zeta }^{\prime }(1-2\varkappa )}{(2\varkappa -1)!}{x}^{2\varkappa -1}\hfill \\ \hfill & =-\sum _{j=1}^{\varkappa -1}\frac{1}{(2\varkappa -2j-1)!}\zeta (-(2\varkappa -2j-1))\zeta (2j+1)x^{2\varkappa -2j-1}+\zeta (2\varkappa +1)x^{-1}\hfill \\ \hfill & -\frac{1}{2}\zeta \left(2\varkappa \right)-\frac{{\zeta }^{\prime }(1-2\varkappa )}{(2\varkappa -1)!}{x}^{2\varkappa -1}\hfill \\ \hfill & =\sum _{j=1}^{\varkappa }\frac{1}{(2\varkappa -2j)!}{B}_{2\varkappa -2j}\zeta (2j+1)x^{2\varkappa -2j-1}-\frac{1}{2}\zeta \left(2\varkappa \right)-\frac{{\zeta }^{\prime }(1-2\varkappa )}{(2\varkappa -1)!}{x}^{2\varkappa -1}.\hfill \end{array}$$

(22)

Corollary 1.

(i) For every even integer α, we have the Ewald expansion

$$\begin{array}{cc}\hfill \sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-nx}=\mathrm{P}\left(x\right)& +2{(-1)}^{\frac{\alpha }{2}}{\left(2\pi \right)}^{-\alpha }{x}^{\alpha -1}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)G_{1,2}^{2,1}\left(\frac{4{\pi }^{2}n}{x}\left|\begin{array}{c}0\\ 0,0\end{array}\right.\right)\hfill \\ \hfill & +2{(-1)}^{\frac{\alpha }{2}}{\left(2\pi \right)}^{-\alpha }{x}^{\alpha -1}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)G_{1,2}^{2,1}\left(-\frac{4{\pi }^{2}n}{x}\left|\begin{array}{c}0\\ 0,0\end{array}\right.\right),\hfill \end{array}$$

(23)

and the (Wigert–Bellman) approximate functional equation

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-xn}\hfill \\ \hfill & ={(-1)}^{\frac{\alpha +1}{2}}{\left(\frac{x}{2\pi }\right)}^{\alpha -1}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-\frac{4{\pi }^2}{x}n}+\mathrm{P}\left(x\right)+O\left(1\right),x=O\left(1\right),\hfill \end{array}$$

(24)

where $P_{2\varkappa }\left(x\right)$ is given by (22), while $P_{-2\varkappa }\left(x\right)$ is given by (15), which reads

$${\mathrm{P}}_{-2\varkappa }\left(x\right)=\Gamma (1+2\varkappa )\zeta (1+2\varkappa )x^{-2\varkappa -1}+\zeta (1-2\varkappa )x^{-1}.$$

(25)

(ii) For every odd integer α, we have the Bochner modular relation (10). In a more symmetric form, it reads

$$\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-2\pi nx}={(-1)}^{\frac{\alpha -1}{2}}{x}^{\alpha -1}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-\frac{2\pi n}{x}}+{\mathrm{P}}_{\alpha }\left(2\pi x\right),$$

(26)

where ${\mathrm{P}}_{2\varkappa +1}\left(2\pi x\right)$ is given by (17). For $\alpha <0$, the residual function ${\mathrm{P}}_{\alpha }\left(x\right)$ is given by (15), and for when $0>\alpha =-(2\varkappa +1)$ is odd, it reduces to

$${\mathrm{P}}_{-2\varkappa -1}\left(x\right)=\Gamma (2\varkappa +2)\zeta (2\varkappa +2)x^{-2\varkappa -2}-\frac{1}{2}\zeta (-2\varkappa -1)-\frac{1}{2}{\delta }_{\varkappa 0}{x}^{-1}.$$

(27)

Thus, (26) may be written as

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-2\pi nx}+\frac{1}{2}\zeta \left(\alpha \right)\hfill \\ \hfill & ={(-1)}^{\frac{\alpha -1}{2}}{x}^{\alpha -1}\left(\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-\frac{2\pi n}{x}}+{(-1)}^{\frac{\alpha -1}{2}}\Gamma (1-\alpha )\zeta (1-\alpha ){\left(2\pi \right)}^{\alpha -1}\right)\hfill \\ \hfill & -\frac{1}{2}{\delta }_{\frac{\alpha -1}{2}0}{\left(2\pi x\right)}^{-1}\hfill \\ \hfill & ={(-1)}^{\frac{\alpha -1}{2}}{x}^{\alpha -1}\left(\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-\frac{2\pi n}{x}}+\frac{1}{2}\zeta \left(\alpha \right)\right)-\frac{1}{2}{\delta }_{\frac{-1-\alpha }{2}0}{\left(2\pi x\right)}^{-1}\hfill \end{array}$$

(28)

which motivates the definition of (34).

The following theorem is a generalization of (Theorem 1, [17]) (which is a consequence of (§3.3, pp. 80–84, [9])).

Theorem 2.

(i) Suppose α is an even integer. Then, the Wigert–Bellman approximate functional Equation (24) gives an approximate automorphy. On the other hand, the Ewald expansion (23) reads

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-nx}\hfill \\ \hfill & =\mathrm{P}\left(x\right)+2{(-1)}^{\varkappa }{\left(2\pi \right)}^{-\alpha }{x}^{\alpha -1}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)\left(\psi \left(1,1;-\frac{4{\pi }^{2}n}{x}\right)+\psi \left(1,1;\frac{4{\pi }^{2}n}{x}\right)\right).\hfill \end{array}$$

(29)

In particular,

$$\begin{array}{cc}\hfill \sum _{n=1}^{\infty }d\left(n\right)e^{-\pi xn}& =\frac{1}{\pi x}(\gamma -log\pi x)+\frac{1}{4}\hfill \\ & +\frac{2}{\pi x}\sum _{n=1}^{\infty }d\left(n\right)\left\{\psi \left(1,1;-\frac{4{\pi }^2}{x}n\right)+\psi \left(1,1;\frac{4{\pi }^2}{x}n\right)\right\},\hfill \end{array}$$

(30)

or

$$\sum _{n=1}^{\infty }d\left(n\right)e^{-xn}=\frac{1}{x}(\gamma -logx)+\frac{1}{4}+O\left(\right|x\left|\right),$$

(31)

whence

$$\underset{z\to 0^{+}}{lim}\left(\sum _{n=1}^{\infty }d\left(n\right)e^{-xn}-\frac{1}{x}(\gamma -logx)\right)=\frac{1}{4}.$$

(32)

(ii) If α is an odd integer, then the functional equation is of type Hecke (52), so that the Bochner modular relation (10) is true by the Hecke correspondence. In the case $\alpha =-(2\varkappa +1)<0$, (10) amounts to the automorphy (38) of the Eisenstein series, whence it may be viewed as the q-expansion (Fourier series) of a modular form. In the case $\alpha =2\varkappa +1>0$, it amounts to Ramanujan’s formula with residual function (17), entailing the eta-transformation formula for $\alpha =1$, which may be viewed as the q-expansion of a modular form of half-integral weight.

Proof. 

This is the special case of Theorem 1 with $k=1$.

In the main case of even $\alpha \ge 0$, we use the identities ((p. 216, (8), p. 264, (2), [11]))

$$\begin{array}{cc}\hfill G_{1,2}^{2,1}\left(x\left|\begin{array}{c}0\\ 0,0\end{array}\right.\right)& =x^{-\frac{1}{2}}{e}^{\frac{1}{2}x}{W}_{-\frac{1}{2},0}\left(x\right)\hfill \\ \hfill & =x^{-\frac{1}{2}}{e}^{\frac{1}{2}x}{e}^{-\frac{1}{2}x}{x}^{-\frac{1}{2}}\psi (1,1;x)\hfill \\ \hfill & =\psi (1,1;x),\hfill \end{array}$$

where $W_{\mu ,\nu }\left(x\right)$ is the Whittaker function (p. 264, (2), [11]), and $\psi (a,c;x)$ is the confluent hypergeometric function ([11]). By

$$\psi (1,1;x)=e^x\Gamma (0,x)=x^{-1}\left(\sum _{m=0}^{M-1}\frac{m!}{{(-x)}^m}+{O\left(\right|x|}^{-M})\right)$$

with $\Gamma (a,x)$ signifying the incomplete gamma function of the second kind (p. 278, (1), p. 226, (21), [11]), we see that each summand on the right side of (30) is estimated by $O\left(\frac{{\left|x\right|}^2}{n^2}\right)$, so that the series on the right side of (30) is absolutely convergent and is of order $O\left(\right|z{|}^2),z\to 0$. The residual function ${\mathrm{P}}_1\left(x\right)$ is given by (21), whence (31).

Other cases of $\alpha$ will be discussed in Section 3. □

The Lambert series with exponential weight contains the previous result in [17]. Research on divisor functions has not been exhausted, especially with respect to the generalized Euler constants studied in [18] and elaborated in [19]. The latter paper also gives the method for expressing the residual function ${\mathrm{P}}_{\alpha }\left(x\right)$ for $k>1$. We hope to return to this elsewhere.

3. $\mathbf{q}$-Expansion and Proof of Theorem 2

Here, we worked with the Bochner modular relations that live in the right half-plane $\mathcal{RHP}=\{x\in \mathbb{C}|\mathrm{Re}x>0\}$. Below,

$$x=-i\tau ,$$

(33)

$\mathcal{RHP}$ is mapped onto the upper half-plane $\mathcal{H}=\{\tau \in \mathbb{C}|\mathrm{Im}\tau >0\}$ and conversely. In this section, we view the above results as Hecke modular relations, i.e., as objects in $\mathcal{H}$, and clarify to what extent the q-expansion aspect is retained, where $q=e^{2\pi i\tau }$. The Fourier expansion or the Laurent expansion at ∞ is of the form ${\sum }_{n=0}^{\infty }{a}_n{e}^{2\pi in\tau }={\sum }_{n=0}^{\infty }{a}_n{q}^n$, and is often referred to as the q-expansion.

Equation (28) motivates the following.

Definition 1.

For every $k\in \mathbb{N}$, Kurokawa introduced the general Eisenstein series

$$E_k\left(\tau \right)=\frac{\zeta (1-k)}{2}+\sum _{n=1}^{\infty }{\sigma }_{k-1}\left(n\right)e^{2\pi in\tau },\tau \in \mathcal{H}.$$

(34)

Kurokawa [20] and Koyama-Kurokawa [21] studied the following limiting values using the Lipschitz summation formula:

$$\underset{\tau \to x}{lim}\left(E_k\left(-\frac{1}{\tau }\right)-{\tau }^k{E}_k\left(\tau \right)\right),\forall x\in \mathbb{R}.$$

(35)

This has been elucidated and generalized by Chapman [22].

In the case that $\alpha$ is odd, (26) leads to

$$\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{2\pi in\tau }={\tau }^{\alpha -1}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-\frac{2\pi in}{\tau }}+\mathrm{P}\left(\frac{2\pi \tau }{i}\right),$$

(36)

which is ((2.11), [17]).

In the case $\alpha =-2\varkappa -1,\varkappa \ge 1$, (28) reads

$$\begin{array}{c}\hfill \sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{2\pi in\tau }+\frac{1}{2}\zeta \left(\alpha \right)={\tau }^{\alpha -1}\left(\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-\frac{2\pi in}{\tau }}+\frac{1}{2}\zeta \left(\alpha \right)\right),\end{array}$$

(37)

or

$$E_{1-\alpha }\left(\tau \right)={\tau }^{\alpha -1}{E}_{1-\alpha }\left(-\frac{1}{\tau }\right),$$

(38)

which establishes the negative odd integer case of Theorem 2, (ii) if we specify a modular form whose q-expansion is (34).

In slightly different notation from (p. 83, [23]), we introduce the Eisenstein series

$$G_{2k}\left(\tau \right):=\sum _{\begin{array}{c}m,n\in \mathbb{Z}\\ (m,n)\ne (0,0)\end{array}}\frac{1}{{(n\tau +m)}^{2k}}.$$

(39)

Lemma 2.

$G_{2k}\left(\tau \right)$ is a modular form of weight $2k$. The Laurent expansion (or q-expansion) reads

$$G_{2k}\left(\tau \right)=2\zeta \left(2k\right)+\frac{2{\left(2\pi i\right)}^{2k}}{(2k-1)!}\sum _{n=1}^{\infty }{\sigma }_{2k-1}\left(n\right)e^{2\pi in\tau },\tau \in \mathcal{H}.$$

(40)

We may also express this as

$$\frac{1}{2\zeta \left(2k\right)}{G}_{2k}\left(\tau \right)=1-\frac{4k}{B_{2k}}\sum _{n=1}^{\infty }{\sigma }_{2k-1}\left(n\right)e^{2\pi in\tau }=E_{2k}\left(\tau \right),$$

(41)

where $E_{2k}\left(\tau \right)$ is the (normalized) Eisenstein series of weight $2k$ ((34), p. 92, [23]).

Proof. 

The first assertion is (Proposition 4, p. 83, [23]); (40) is given on (p. 92, [23]) and (41) is given by (16). □

Hence, the even-suffix case of Definition 1, $E_{2k}\left(\tau \right)=\frac{\zeta (1-2k)}{4\zeta \left(2k\right)}{G}_{2k}\left(\tau \right)$, is the modular form whose q-expansion is (34).

We turn to the positive odd integer case of $\alpha$. The residual function is given by (18), and we are to distribute each half of it to both sides and make (37) symmetric. Such an endeavor was made by Toyoizumi [24] and elucidated as a modular relation [9]. Then, (Theorem 3, [24]) gives a generalized Hecke correspondence. In this theorem, the parameter b in (52) must be odd if $b>0$. But in the present case, $b=1-\alpha <0$ is even, and the theorem does not apply. The negative odd integer case is contained in (Theorem 3, [24]).

We turn to the case $\alpha =1$ and note that between the Riemann zeta-function and the exp-function method (a technique in solving nonlinear partial differential equations), the readers can see a connection with the heat kernel and modular relations in [25] and some recent references in [26].

The Dedekind eta-function is defined by

$$\eta \left(\tau \right)=e^{\frac{\pi iz}{12}}\prod _{n=1}^{\infty }(1-e^{2\pi in\tau }),\mathrm{Im}\tau >0.$$

(42)

Lemma 3.

The discriminant function $\Delta \left(\tau \right)$, which is the 24th power of the Dedekind eta-function, is a cusp form of weight 12. And a fortrori, $\eta \left(\tau \right)$, is a cusp form of weight $\frac{1}{2}$.

Proof can be found, e.g., in (pp. 69–70, [27]).

We prove that $\eta \left(\tau \right)$ satisfies the transformation formula (cf. [28]).

$$\eta \left(-\frac{1}{\tau }\right)=\sqrt{\frac{\tau }{i}}\eta \left(\tau \right)$$

(43)

or

$$log\eta \left(-\frac{1}{\tau }\right)=log\eta \left(\tau \right)+2\pi i\frac{\tau }{i}.$$

(44)

Let

$$F\left(\tau \right)=\sum _{n=1}^{\infty }{\sigma }_{-1}\left(n\right)q^n=\sum _{l,m=1}^{\infty }\frac{1}{l}{q}^{lm}=-\sum _{k=1}^{\infty }log(1-q^k).$$

(45)

Then,

$$F\left(\tau \right)=\frac{\pi i}{12}\tau -log\eta \left(\tau \right).$$

(46)

Here, (36) with $\alpha >0$ is odd, and (26) leads to

$$\begin{array}{c}\hfill F\left(\tau \right)-\frac{\pi i}{12}\tau =F\left(-\frac{1}{\tau }\right)-\frac{\pi i}{12}\left(-\frac{i}{\tau }\right)+\frac{1}{2}log\frac{\tau }{i},\end{array}$$

(47)

which is the formula deduced by A. Weil (p. 401, [29]). Using (46), we deduce (44).

Finally we treat the case of $\alpha <0$ being an even integer. We analyze the approximate automorphy (24), where the residual function is given by (25). Equation (24) reads

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-2\pi xn}\hfill \\ \hfill & ={(-1)}^{\frac{\alpha +1}{2}}{x}^{\alpha -1}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-\frac{2\pi }{x}n}+\mathrm{P}\left(2\pi x\right)+O\left(1\right),x=O\left(1\right),\hfill \end{array}$$

or

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{2\pi i\tau n}\hfill \\ \hfill & =-{\tau }^{\alpha -1}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-\frac{2\pi i}{\tau }n}+\mathrm{P}(-2\pi i\tau )+O\left(1\right),x=O\left(1\right)\hfill \end{array}$$

(48)

where (25) reads

$${\mathrm{P}}_{-2\varkappa }(-2\pi i\tau )=-\Gamma (1-\alpha )\zeta (1-\alpha )\left(2\pi i\right){\tau }^{\alpha -1}+\zeta (1+\alpha )\frac{2\pi i}{\tau }.$$

(49)

Here, the argument breaks down since we cannot distribute the terms in (49) as in (47); the constant term is contained in the error term $O\left(1\right)$.

We do not dwell on the case when $\alpha >0$ is even, where the residual function is given by (22).

4. Proof of Theorem 1

Since Theorem 1 is the Bochner modular relation corresponding to the functional equation for $\phi \left(s\right)$, the following lemma is crucial.

Lemma 4.

Zeta-function $\phi \left(s\right)$ (3) satisfies the asymmetric functional equation

$$\phi (1-s)={\left(4{\left(2\pi \right)}^{-2s+\alpha }cos\frac{\pi }{2}scos\frac{\pi }{2}(s-\alpha )\Gamma \left(s\right)\Gamma (s-\alpha )\right)}^k\phi (s-\alpha ).$$

(50)

If α is an odd integer, then (50) amounts to

$$\begin{array}{c}\hfill \Gamma {(1-s)}^k\phi (1-s)={\left({\left(2\pi \right)}^{-2s+\alpha +1}{(-1)}^{\frac{\alpha -1}{2}}\Gamma (\alpha +1-s)\right)}^k\phi (s-\alpha ).\end{array}$$

(51)

For $k=1$, this reduces to the Hecke functional equation

$${\left(2\pi \right)}^{-s}\Gamma \left(s\right)\phi \left(s\right)={(-1)}^{\frac{b}{2}}{\left(2\pi \right)}^{b-s}\Gamma (b-s)\phi (b-s),b=1-\alpha .$$

(52)

If α is an even integer, then we transform (50) into (58) or (60) as the case may be.

Proof. 

According to the parity of $\alpha$, we apply

$$cos\frac{\pi }{2}(s-\alpha )=\left\{\begin{array}{cc}sin\frac{\pi }{2}ssin\frac{\pi }{2}\alpha ,\hfill & \alpha =2\varkappa +1,\hfill \\ cos\frac{\pi }{2}scos\frac{\pi }{2}\alpha ,\hfill & \alpha =2\varkappa ,\hfill \end{array}\right.$$

whence the product of cosines in (50) amounts to

$$\left\{\begin{array}{cc}\frac{1}{2}{(-1)}^{\varkappa }sin\pi s,\hfill & \alpha =2\varkappa +1,\hfill \\ {(-1)}^{\varkappa }{cos}^2\frac{\pi }{2}s,\hfill & \alpha =2\varkappa .\hfill \end{array}\right.$$

The proof of the case with an even $\alpha$ is given in the proof of Theorem 1. □

Let ($\mathrm{Re}x>0$)

$$V\left(x\right)=V_k\left(x\right)=\frac{1}{2\pi i}{\int }_{\left(c\right)}\Gamma {\left(s\right)}^k{x}^{-s}\mathrm{d}s=G_{0,k}^{k,0}\left(x\left|\begin{array}{c}-\\ \underset{k}{\underbrace{0,···,0}}\end{array}\right.\right),$$

(53)

be the Voronoĭ function, cf. ((3.2), [4]), (p. 62, [9]), [30,31], (p. 347, [32]), [33,34], etc. We apply the Voronoĭ transform to deduce the correct form of ((3.3), [4])

$$\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)V\left(nx\right)=\frac{1}{2\pi i}{\int }_{\left(c\right)}\Gamma {\left(s\right)}^k\phi \left(s\right)x^{-s}\mathrm{d}s,$$

(54)

where $c>{\sigma }_{\phi }$, and $\left(c\right)$ indicates the Bromwich contour $\sigma =c,-\infty <t<\infty$.

For $k=1$, (54) is the Hecke gamma transform:

$$\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-nx}=\frac{1}{2\pi i}{\int }_{\left(c\right)}\Gamma \left(s\right)\phi \left(s\right)x^{-s}\mathrm{d}s.$$

(55)

Proof of Theorem 1.

Following a standard procedure of moving the line to the left up to $\left(d\right)$, where $-1<d+\alpha <0$ ($d=-\alpha -\frac{1}{2}$, say), whereby noting that the horizontal integrals vanish in the limit as $\left|t\right|\to \infty$, (54) becomes

$$\frac{1}{2\pi i}{\int }_{\left(c\right)}\Gamma {\left(s\right)}^k\phi \left(s\right)x^{-s}\mathrm{d}s=\frac{1}{2\pi i}{\int }_{\left(d\right)}\Gamma {\left(s\right)}^k\phi \left(s\right)x^{-s}\mathrm{d}s+\mathrm{P}\left(x\right).$$

(56)

Here, $\mathrm{P}\left(x\right)={\mathrm{P}}_{\alpha }\left(x\right)$ is the residual function consisting of the sum of residues of the integrand $\Gamma {\left(s\right)}^k\phi \left(s\right)x^{-s}$ at $-\alpha ,1-\alpha ,···,-1,0,1$ for $\alpha >0$ and at $0,1,1-\alpha$ for $\alpha \le 0$. Writing $1-s$ for s in (56), we see that the right-hand side of (56) becomes $\frac{1}{2\pi i}{\int }_{(1-d)}\Gamma {(1-s)}^k\phi (1-s)x^{s-1}\mathrm{d}s+\mathrm{P}\left(x\right)$. Hence,

$$\begin{array}{c}\hfill \sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)V\left(nx\right)=\frac{1}{x}\frac{1}{2\pi i}{\int }_{(1-d)}\Gamma {(1-s)}^k\phi (1-s)x^s\mathrm{d}s+\mathrm{P}\left(x\right).\end{array}$$

(57)

I. In the case of $\alpha$ being even, we deduce from (50) that

$$\begin{array}{cc}\hfill & \Gamma {(1-s)}^k\phi (1-s)\hfill \\ \hfill & ={\left(4{\left(2\pi \right)}^{-2s+\alpha }{(-1)}^{\frac{\alpha }{2}}{\left(cos\frac{\pi }{2}s\right)}^2\Gamma (1-s)\Gamma \left(s\right)\Gamma (s-\alpha )\Gamma (\alpha +1-s)\right)}^k\phi (s-\alpha )\hfill \\ \hfill & ={\left(4{\left(2\pi \right)}^{-2s+\alpha }{(-1)}^{\frac{\alpha }{2}}{\left(cos\frac{\pi }{2}s\right)}^2\Gamma {(s-\alpha )}^2\Gamma (\alpha +1-s)\right)}^k\phi (s-\alpha )\hfill \end{array}$$

on viewing $\frac{\pi }{sin\pi s}=\frac{\pi }{sin\pi (s-\alpha )}$. Hence we, obtain

$$\begin{array}{cc}\hfill & \Gamma {(1-s)}^k\phi (1-s)\hfill \\ \hfill & =\sum _{j=0}^{2k}\left(\genfrac{}{}{0pt}{}{2k}{j}\right)e^{-\pi i(k-j)s}{\left({\left(2\pi \right)}^{-2s+\alpha }{(-1)}^{\frac{\alpha }{2}}\Gamma {(s-\alpha )}^2\Gamma (\alpha +1-s)\right)}^k\phi (s-\alpha )\hfill \\ \hfill & ={\left({\left(2\pi \right)}^{\alpha }{(-1)}^{\frac{\alpha }{2}}\right)}^k\sum _{j=0}^{2k}\left(\genfrac{}{}{0pt}{}{2k}{j}\right)\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)n^{\alpha }\Gamma {(s-\alpha )}^{2k}\Gamma {(\alpha +1-s)}^k{\left({\left(4{\pi }^2\right)}^k{e}^{-\pi i(k-j)}n\right)}^{-s},\hfill \end{array}$$

(58)

where the expansion holds for $\sigma >\alpha +1$.

Substituting (58) into (63), we obtain

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)V\left(nx\right)=\frac{1}{2\pi i}{\int }_{(1-d)}\Gamma {(1-s)}^k\phi (1-s)x^{s-1}\mathrm{d}s+\mathrm{P}\left(x\right)\hfill \\ \hfill & =\frac{{\left({\left(2\pi \right)}^{\alpha }{(-1)}^{\frac{\alpha }{2}}\right)}^k}{x}\sum _{j=0}^{2k}\left(\genfrac{}{}{0pt}{}{2k}{j}\right)\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)n^{\alpha }\hfill \\ \hfill & \times \frac{1}{2\pi i}{\int }_{(1-d)}\Gamma {(s-\alpha )}^{2k}\Gamma {(\alpha +1-s)}^k{\left(\frac{{\left(4{\pi }^2\right)}^k{e}^{-\pi i(k-j)}}{x}n\right)}^s\mathrm{d}s+\mathrm{P}\left(x\right)\hfill \\ \hfill & =\frac{{\left({\left(2\pi \right)}^{\alpha }{(-1)}^{\frac{\alpha }{2}}\right)}^k}{x}\sum _{j=0}^{2k}\left(\genfrac{}{}{0pt}{}{2k}{j}\right)\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)n^{\alpha }\hfill \\ \hfill & \times G_{k,2k}^{2k,k}\left(\frac{{\left(4{\pi }^2\right)}^k{e}^{-\pi i(k-j)}}{x}n\left|\begin{array}{c}\stackrel{k}{\overbrace{-\alpha ,···,-\alpha }}\hfill \\ \underset{2k}{\underbrace{-\alpha ,···,-\alpha }}\hfill \end{array}\right.\right)+\mathrm{P}\left(x\right)\hfill \\ \hfill & =\frac{{\left({\left(2\pi \right)}^{\alpha }{(-1)}^{\frac{\alpha }{2}}\right)}^k}{x}\sum _{j=0}^{2k}\left(\genfrac{}{}{0pt}{}{2k}{j}\right)\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)n^{\alpha }{\left(\frac{{\left(4{\pi }^2\right)}^k{e}^{-\pi i(k-j)}}{x}n\right)}^{-\alpha }\hfill \\ \hfill & \times G_{k,2k}^{2k,k}\left(\frac{{\left(4{\pi }^2\right)}^k{e}^{-\pi i(k-j)}}{x}n\left|\begin{array}{c}\stackrel{k}{\overbrace{0,···,0}}\hfill \\ \underset{2k}{\underbrace{0,···,0}}\hfill \end{array}\right.\right)+\mathrm{P}\left(x\right),\hfill \end{array}$$

(59)

which proves (5).

I’. Bellman applies in place of (58):

$$\begin{array}{cc}\hfill \Gamma {(1-s)}^k\phi (1-s)& ={\left({\left(2\pi \right)}^{-2s+\alpha +1}{(-1)}^{\frac{\alpha }{2}}\left(cot\frac{\pi }{2}s\right)\Gamma (s-\alpha )\right)}^k\phi (s-\alpha )\hfill \\ \hfill & ={\left({\left(2\pi \right)}^{\alpha +1}{(-1)}^{\frac{\alpha }{2}}\right)}^k\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)n^{\alpha }{\left(cot\frac{\pi }{2}s\right)}^k\Gamma {(s-\alpha )}^k{\left({\left(4{\pi }^2\right)}^{k}n\right)}^{-s}.\hfill \end{array}$$

(60)

Substituting (60) into (57), we obtain

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)V\left(nx\right)=\frac{1}{2\pi i}{\int }_{(1-d)}\Gamma {(1-s)}^k\phi (1-s)x^{s-1}\mathrm{d}s+\mathrm{P}\left(x\right)\hfill \\ \hfill & =\frac{{\left({\left(2\pi \right)}^{\alpha +1}{(-1)}^{\frac{\alpha }{2}}\right)}^k}{x}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)n^{\alpha }\frac{1}{2\pi i}{\int }_{(1-d)}{\left(cot\frac{\pi }{2}s\right)}^k\Gamma {(s-\alpha )}^k{\left(\frac{{\left(4{\pi }^2\right)}^k}{x}n\right)}^{-s}\mathrm{d}s\hfill \\ \hfill & +\mathrm{P}\left(x\right),\hfill \end{array}$$

(61)

of which the $\alpha =0$ case coincides with the first formula in (p. 551, [4]). Now, we follow Bellman’s method of approximating ${\left(cot\frac{\pi }{2}s\right)}^k$ using $i^k+O\left(e^{-\pi \left|t\right|}\right)$, $\left|t\right|\to \infty$.

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)V\left(nx\right)=\frac{1}{2\pi i}{\int }_{(1-d)}\Gamma {(1-s)}^k\phi (1-s)x^{s-1}\mathrm{d}s+\mathrm{P}\left(x\right)\hfill \\ \hfill & =\frac{{\left({\left(2\pi \right)}^{\alpha +1}{(-1)}^{\frac{\alpha +1}{2}}\right)}^k}{x}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)n^{\alpha }{\left(\frac{{\left(4{\pi }^2\right)}^k}{x}n\right)}^{-\alpha }V\left(\frac{{\left(4{\pi }^2\right)}^k}{x}n\right)\hfill \\ \hfill & +\frac{{\left({\left(2\pi \right)}^{\alpha +1}{(-1)}^{\frac{\alpha +1}{2}}\right)}^k}{x}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)n^{\alpha }\hfill \\ \hfill & \times \frac{1}{2\pi i}{\int }_{(1-d)}\left({\left(cot\frac{\pi }{2}s\right)}^k-i^k\right)\Gamma {(s-\alpha )}^k{\left(\frac{{\left(4{\pi }^2\right)}^k}{x}n\right)}^{-s}\mathrm{d}s+\mathrm{P}\left(x\right)\hfill \\ \hfill & ={(-1)}^{\frac{\alpha +1}{2}k}{\left(\frac{x}{{\left(2\pi \right)}^k}\right)}^{\alpha -1}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)V\left(\frac{{\left(4{\pi }^2\right)}^k}{x}n\right)+\mathrm{P}\left(x\right)+{O\left(\right|x|}^{-d}),\hfill \end{array}$$

(62)

which leads to (6).

II. In the case of $\alpha$ being odd, from (50), we obtain

$$\begin{array}{cc}\hfill \Gamma {(1-s)}^k\phi (1-s)& =\Gamma {(1-s)}^k{\left(2{\left(2\pi \right)}^{-2s+\alpha }{(-1)}^{\frac{\alpha -1}{2}}sin\pi s\Gamma \left(s\right)\Gamma (\alpha +1-s)\right)}^k\phi (s-\alpha )\hfill \\ \hfill & =\Gamma {(1-s)}^k{\left(2{\left(2\pi \right)}^{-2s+\alpha }{(-1)}^{\frac{\alpha -1}{2}}\frac{\pi }{\Gamma (1-s)}\Gamma (\alpha +1-s)\right)}^k\phi (s-\alpha )\hfill \\ \hfill & ={\left({\left(2\pi \right)}^{-2s+\alpha +1}{(-1)}^{\frac{\alpha -1}{2}}\Gamma (\alpha +1-s)\right)}^k\phi (s-\alpha )\hfill \\ \hfill & ={\left({\left(2\pi \right)}^{\alpha +1}{(-1)}^{\frac{\alpha -1}{2}}\right)}^k\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)n^{\alpha }\Gamma {(\alpha +1-s)}^k{\left({\left(4{\pi }^2\right)}^{k}n\right)}^{-s}.\hfill \end{array}$$

(63)

Substituting (63) into (57), we obtain

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)V\left(nx\right)=\frac{1}{2\pi i}{\int }_{(1-d)}\Gamma {(1-s)}^k\phi (1-s)x^{s-1}\mathrm{d}s+\mathrm{P}\left(x\right)\hfill \\ \hfill & =\frac{{\left({\left(2\pi \right)}^{\alpha +1}{(-1)}^{\frac{\alpha -1}{2}}\right)}^k}{x}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)n^{\alpha }\frac{1}{2\pi i}{\int }_{(1-d)}\Gamma {(\alpha +1-s)}^k{\left(\frac{{\left(4{\pi }^2\right)}^k}{x}n\right)}^{-s}\mathrm{d}s+\mathrm{P}\left(x\right)\hfill \\ \hfill & =\frac{{\left({\left(2\pi \right)}^{\alpha +1}{(-1)}^{\frac{\alpha -1}{2}}\right)}^k}{x}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)n^{\alpha }{\left(\frac{{\left(4{\pi }^2\right)}^k}{x}n\right)}^{-\alpha }{G}_{0,k}^{k,0}\left(\frac{{\left(4{\pi }^2\right)}^k}{x}n\left|\begin{array}{c}-\\ \underset{k}{\underbrace{0,···,0}}\end{array}\right.\right)+\mathrm{P}\left(x\right)\hfill \\ \hfill & =\frac{{\left({\left(2\pi \right)}^{\alpha +1}{(-1)}^{\frac{\alpha -1}{2}}\right)}^k}{x}{\left(\frac{{\left(4{\pi }^2\right)}^k}{x}\right)}^{-\alpha }\sum _{n=1}^{\infty }{\sigma }_{-\alpha }^k\left(n\right)V\left(\frac{{\left(4{\pi }^2\right)}^k}{x}n\right)+\mathrm{P}\left(x\right),\hfill \\ \hfill \end{array}$$

(64)

which proves (7).

In the case of $k=1$, (64) amounts to (37). □

5. Conclusions

As is alluded to in the Abstract, we study the product of two Riemann zeta-functions with the difference $\alpha$ in the variables. Classically, the odd-difference case has been studied, and here, we investigate the even-difference case. It turns out that the obtained results are much more involved, but one could still extract many useful results.