On Discrete Shifts of Some Beurling Zeta Functions

Abstract

We consider the Beurling zeta function ${\zeta }_{\mathbb{P}}\left(s\right)$, $s=\sigma +it$, of the system of generalized prime numbers $\mathbb{P}$ with generalized integers m satisfying the condition ${\sum }_{m⩽x}1=ax+O\left(x^{\delta }\right)$, $a>0$, $0⩽\delta <1$, and suppose that ${\zeta }_{\mathbb{P}}\left(s\right)$ has a bounded mean square for $\sigma >{\sigma }_{\mathbb{P}}$ with some ${\sigma }_{\mathbb{P}}<1$. Then, we prove that, for every $h>0$, there exists a closed non-empty set of analytic functions that are approximated by discrete shifts ${\zeta }_{\mathbb{P}}(s+ilh)$. This set shifts has a positive density. For the proof, a weak convergence of probability measures in the space of analytic functions is applied.

1. Introduction

Let $\mathbb{P}$ be a system of generalized prime numbers $1<p_1⩽p_2⩽p_3⩽\cdots ⩽p_n⩽\cdots$, ${lim}_{n\to \infty }{p}_n=+\infty$, and ${\mathbb{N}}_{\mathbb{P}}$ the corresponding system of generalized integers

$$m=p_1^{{\alpha }_1}\cdots p_r^{{\alpha }_r},{\alpha }_k\in \mathbb{N}\cup \left\{0\right\}={\mathbb{N}}_0,k=1,2,\dots ,r,r\in \mathbb{N}.$$

(1)

If identical generalized integers have different representations (1), they are considered with multiplicities. Generalized primes and integers were introduced by Beurling in [1]. For investigation of the distribution of generalized primes, he also defined the zeta function

$${\zeta }_{\mathbb{P}}\left(s\right)=\underset{m\in {\mathbb{N}}_{\mathbb{P}}}{\sum _{m=1}^{\infty }}{\displaystyle \frac{1}{m^s}},\mathrm{or}{\zeta }_{\mathbb{P}}\left(s\right)=\prod _{p\in \mathbb{P}}{\left(1-{\displaystyle \frac{1}{p^s}}\right)}^{-1},s=\sigma +it,\sigma >{\sigma }_{\mathbb{P}},$$

(2)

which is generalization of the classical Riemann zeta function

$$\zeta \left(s\right)=\sum _{m=1}^{\infty }{\displaystyle \frac{1}{m^s}}=\prod _q{\left(1-{\displaystyle \frac{1}{q^s}}\right)}^{-1},\sigma >1,$$

where the product is taken over rational prime numbers q. We observe that generalized primes and generalized integers can be identical numbers; thus, in (2), the sum and the product are taken by counting multiplicities of $m\in {\mathbb{N}}_{\mathbb{P}}$ and $p\in \mathbb{P}$. Moreover, the half-plane $\sigma >{\sigma }_{\mathbb{P}}$ depends on the system $\mathbb{P}$.

The function $\zeta \left(s\right)$ is the principal analytic tool for the investigation of the distribution of prime numbers, i.e., for the proof of the asymptotic formula for the function

$$\pi \left(x\right)=\sum _{q⩽x}1,x\to \infty .$$

From the fundamental works of B. Riemann [2], C. de la Valeé Poussin [3,4,5] and J. Hadamard [6], it is known that

$$\pi \left(x\right)=\underset{2}{\overset{x}{\int }}{\displaystyle \frac{\mathrm{d}v}{logv}}(1+o\left(1\right)),x\to \infty .$$

(3)

The function $\zeta \left(s\right)$ is meromorphically continued, with a simple pole at $s=1$ and residue 1. The study of $\pi \left(x\right)$ is closely connected to the location of non-trivial zeros of $\zeta \left(s\right)$ lying in the strip $\{s\in \mathbb{C}:0⩽\sigma ⩽1\}$. For example, Formula (3) follows from non-vanishing $\zeta \left(s\right)$ on the line $\sigma =1$. Estimations of the error term in (3) depend on zero free regions for $\zeta \left(s\right)$ on the left of the half-plane $\sigma ⩾1$. If the Riemann hypothesis is true (all non-trivial zeros lie on the line $\sigma =1/2$), then it is known that [7]

$$\pi \left(x\right)=\underset{2}{\overset{x}{\int }}{\displaystyle \frac{\mathrm{d}v}{logv}}+O(x^{1/2}logx),x\to \infty .$$

The best known unconditional result is as follows (see [8]),

$$\pi \left(x\right)=\underset{2}{\overset{x}{\int }}{\displaystyle \frac{\mathrm{d}v}{logv}}+O\left(xexp\left\{-c{\displaystyle \frac{{(logx)}^{3/5}}{{(loglogx)}^{1/5}}}\right\}\right),c>0.$$

The case of generalized primes is more complicated, and applications of the function ${\zeta }_{\mathbb{P}}\left(s\right)$, called the Beurling zeta function, are connected to investigation of the functions

$${\pi }_{\mathbb{P}}\left(x\right)=\sum _{p⩽x}1$$

and

$$N_{\mathbb{P}}\left(x\right)=\underset{m\in {\mathbb{N}}_{\mathbb{P}}}{\sum _{m⩽x}}1$$

as $x\to \infty$.

The analytic extension of the function ${\zeta }_{\mathbb{P}}\left(s\right)$ is closely connected to the asymptotic behavior of $N_{\mathbb{P}}\left(x\right)$. We will use the convenient notation $A{\ll }_{\theta }B$, $A,B\in \mathbb{C}$, $B>0$, which is equivalent to the symbol $O(\dots )$, with the implied constant depending on $\theta$. It is obvious that if $N_{\mathbb{P}}\left(x\right){\ll }_{\mathbb{P}}x$, then ${\sigma }_{\mathbb{P}}=1$. Hence, ${\zeta }_{\mathbb{P}}\left(s\right)$ is analytic in the half-plane $\sigma >1$. Suppose that

$$N_{\mathbb{P}}\left(x\right)-ax{\ll }_{\mathbb{P}}{x}^{\delta },a>0,0⩽\delta <1.$$

(4)

Then, it is easily seen that the Beurling zeta function ${\zeta }_{\mathbb{P}}\left(s\right)$ is analytically extended to the region $\sigma >\delta$, except for the unique simple pole at $s=1$, with ${\mathrm{Res}}_{s=1}{\zeta }_{\mathbb{P}}\left(s\right)=a$.

An old result of E. Landau [9] regarding the distribution of prime ideals, implies that from (4), the estimate

$${\pi }_{\mathbb{P}}\left(x\right)-\underset{2}{\overset{x}{\int }}{\displaystyle \frac{\mathrm{d}v}{logv}}{\ll }_{\mathbb{P}}xexp\left\{-c\sqrt{logx}\right\},c>0,$$

follows.

A strong result of Beurling [1] states that the estimate

$$N_{\mathbb{P}}\left(x\right)-ax{\ll }_{\mathbb{P}}x{\left(logx\right)}^{-\delta },\delta >{\displaystyle \frac{3}{2}},$$

which implies the asymptotics

$${\pi }_{\mathbb{P}}\left(x\right)={\displaystyle \frac{x}{logx}}(1+o\left(1\right)),x\to \infty .$$

Moreover, the latter formula is not true for all systems $\mathbb{P}$ if $\delta =3/2$. On the other hand, H.G. Diamond proved [10] that the estimate

$${\pi }_{\mathbb{P}}\left(x\right)-{\displaystyle \frac{x}{logx}}\ll x{(logx)}^{-\delta },\delta >1,$$

implies

$$N_{\mathbb{P}}\left(x\right)=ax(1+o\left(1\right)),x\to \infty .$$

Interesting results of [11] connect the estimates

$$N_{\mathbb{P}}\left(x\right)-ax\ll x{(logx)}^{-{\delta }_1},{\delta }_1>0,\mathrm{and}{\pi }_{\mathbb{P}}\left(x\right)-\underset{2}{\overset{x}{\int }}{\displaystyle \frac{\mathrm{d}v}{logv}}\ll x{(logx)}^{-{\delta }_2},\forall {\delta }_2>0,$$

and the works [12,13,14] deal with

$$N_{\mathbb{P}}\left(x\right)-ax\ll xexp\left\{-c_1{(logx)}^{{\delta }_1}\right\},{\delta }_1>0,c_1>0,$$

and

$${\pi }_{\mathbb{P}}\left(x\right)-\underset{2}{\overset{x}{\int }}{\displaystyle \frac{\mathrm{d}v}{logv}}\ll xexp\left\{-c_2{(logx)}^{{\delta }_2}\right\},{\delta }_2>0,c_2>0,$$

and ${\delta }_2$ depends on ${\delta }_1$.

In [15], for the analytic extension of ${\zeta }_{\mathbb{P}}\left(s\right)$, the von Mangoldt function

$${\Lambda }_{\mathbb{P}}\left(m\right)=\left\{\begin{array}{cc}logp\hfill & \mathrm{if}m=p^k,k\in \mathbb{N},\hfill \\ 0\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

and the Chebyshev-type function

$${\psi }_{\mathbb{P}}\left(x\right)=\underset{m\in {\mathbb{N}}_{\mathbb{P}}}{\sum _{m⩽x}}\Lambda \left(m\right)$$

were applied.

Theorem 1

([15]). Suppose that, for $0⩽\delta <1$,

$${\psi }_{\mathbb{P}}\left(x\right)-x\ll x^{\delta +\epsilon },\forall \epsilon >0.$$

(5)

Then, the function ${\zeta }_{\mathbb{P}}\left(s\right)$ is analytically continued to the half-plane $\sigma >\delta$, except for unique simple pole at the point $s=1$, and ${\zeta }_{\mathbb{P}}\left(s\right)\ne 0$ in this half-plane.

Under a certain additional condition for the logarithmic derivative of ${\zeta }_{\mathbb{P}}\left(s\right)$, the estimate (5) is also necessary for the analytic extension ${\zeta }_{\mathbb{P}}\left(s\right)$ to the above region.

We observe that the circle of problems considered in [15] is wide, and is not limited by Theorem 1. The authors propose another way involving Mellin transforms. Also, the problem of the functional equation for ${\zeta }_{\mathbb{P}}\left(s\right)$ is discussed, and a criterion for the existence of such an equation is provided. Moreover, the application of the Beurling zeta function in fractal string theory and other potential applications, for example, for investigations of the quasicrystal structure, are also considered.

The study of generalized prime systems and the corresponding Beurling zeta functions is continued. For example, we mention the paper [16] containing various zero-distribution results for ${\zeta }_{\mathbb{P}}\left(s\right)$.

In [17], we started with an approximation of the analytic functions by shifts ${\zeta }_{\mathbb{P}}(s+i\tau )$, $\tau \in \mathbb{R}$. The latter property of zeta functions to approximate entire classes of analytic functions is called universality, and, for the Riemann zeta function $\zeta \left(s\right)$ was found by S.M. Voronin [18]. Later, this property was extended to other zeta functions and classes of the Dirichlet series. For the results and methods, we recommend the informative survey paper of K. Matsumoto [19], as well as the books [20,21,22], and dissertations [23,24]. We recall the last version of Voronin’s universality theorem for $\zeta \left(s\right)$. For this, we introduce the following notation. Let $1/2⩽\kappa <1$, ${\mathcal{K}}_{\kappa }$ denote the set of all compact subsets of the strip $D_{\kappa }=\{s\in \mathbb{C}:\kappa <\sigma <1\}$ with a connected complements, and $H_{\kappa }^{\circ }\left(K\right)$ with $K\in {\mathcal{K}}_{\kappa }$ the set of continuous non-vanishing functions on K that are analytic in the interior of K. Moreover, let

$$m_T(\dots )={\displaystyle \frac{1}{T}}{\mu }_L\{\tau \in [0,T]:\dots \},$$

where ${\mu }_L$ is the Lebesgue measure of the subset of $[0,T]$ satisfying the conditions written in place of dots. Then, the following statement is known.

Theorem 2.

Let $K\in {\mathcal{K}}_{1/2}$ and $g\left(s\right)\in H_{1/2}^{\circ }\left(K\right)$. Then, for every $\epsilon >0$,

$$\underset{T\to \infty }{lim\; inf}{m}_T\left(\underset{s\in K}{sup}|g\left(s\right)-\zeta (s+i\tau )|<\epsilon \right)>0.$$

Moreover,

$$\underset{T\to \infty }{lim}{m}_T\left(\underset{s\in K}{sup}|g\left(s\right)-\zeta (s+i\tau )|<\epsilon \right)$$

exists and is positive for all but at most countably many $\epsilon >0$.

The first assertion of the theorem can be found in [19,20,21,23,24], while the second was proven in [25,26].

Universality theorems for zeta functions have various theoretical and practical applications, including their functional independence, zero-distribution, various estimates, and a description of the behavior of particles in quantum mechanics [27,28,29,30,31,32,33]. Therefore, it is important to extend the set of universal zeta functions and improve the universality theorems. It is known by the Linnik–Ibragimov conjecture that all functions in some half-plane given by the Dirichlet series, having analytic continuation to the left of that half-plane and satisfying certain natural growth conditions are universal in Voronin’s sense (see Section 1.6 of [21]). For this, Beurling zeta functions are a suitable general object. In [17], under restriction (4), one result regarding the approximation of the analytic functions by shifts ${\zeta }_{\mathbb{P}}(s+i\tau )$ has been obtained. The bound for the mean square

$$M_{\sigma }\left(T\right)={\displaystyle \frac{1}{T}}\underset{0}{\overset{T}{\int }}{\left|{\zeta }_{\mathbb{P}}(s+it)\right|}^2\mathrm{d}t$$

is involved in its statement for the definition of the approximation strip $D_{\kappa }$. Denote by $\widehat{\sigma }$ the infimum of $\sigma$ satisfying $M_{\sigma }\left(T\right){\ll }_{\sigma }1$ for all $T>0$, and let $\delta$ be from (4). Set ${\sigma }_{\mathbb{P}}=max(\widehat{\sigma },\delta )$, and suppose that ${\sigma }_{\mathbb{P}}<1$. Then, in [17], the following statement regarding the good approximation properties of the function ${\zeta }_{\mathbb{P}}\left(s\right)$ were obtained. Let $\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)$ be the space of analytic functions on $D_{{\sigma }_{\mathbb{P}}}$ equipped with the topology of uniform convergence on the compact sets.

Theorem 3

([17]). Suppose that the system of generalized primes $\mathbb{P}$ satisfies the estimate (4). Then, there is a non-empty closed subset ${\mathcal{F}}_{\mathbb{P}}\subset \mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)$, such that, for all compact sets $K\in D_{{\sigma }_{\mathbb{P}}}$, $g\left(s\right)\in {\mathcal{F}}_{\mathbb{P}}$ and $\epsilon >0$,

$$\underset{T\to \infty }{lim\; inf}{m}_T\left(\underset{s\in K}{sup}\left|g\left(s\right)-{\zeta }_{\mathbb{P}}(s+i\tau )\right|<\epsilon \right)>0,$$

and the limit

$$\underset{T\to \infty }{lim}{m}_T\left(\underset{s\in K}{sup}\left|g\left(s\right)-{\zeta }_{\mathbb{P}}(s+i\tau )\right|<\epsilon \right)$$

exists and is positive for all but at most countably many $\epsilon >0$.

Theorem 3 asserts that the set of approximating shifts ${\zeta }_{\mathbb{P}}(s+i\tau )$ for every function $g\left(s\right)$ from the set ${\mathcal{F}}_{\mathbb{P}}$ is infinite, it has a positive lower density, and even positive density, except, possibly, for some narrow set of values $\epsilon >0$.

In Theorem 3, analytic functions are approximated by continuous shifts ${\zeta }_{\mathbb{P}}(s+i\tau )$, where $\tau$ can take arbitrary real values from the interval $[0,T]$. There is an another, more convenient, way of using discrete shifts when $\tau$ varies in a certain discrete set. The simplest discrete set is of the form $\{lh:l\in {\mathbb{N}}_0\}$ with fixed $h>0$.

The aim of this paper is a discrete version of Theorem 3. Denote by $\#A$ the cardinality of the set $A\subset \mathbb{R}$, and let

$$m_N(\dots )={\displaystyle \frac{1}{N+1}}\#\{0⩽l⩽N:\dots \},$$

where in place of dots, a condition satisfied by l is to be written, and N runs over the set ${\mathbb{N}}_0$. Now, we state the main result of the paper.

Theorem 4.

Suppose that the system $\mathbb{P}$ satisfies the estimate (4), and $h>0$. Then, there exists a non-empty closed subset ${\mathcal{F}}_{\mathbb{P},h}\subset \mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)$, such that, for all compact sets, $K\in D_{{\sigma }_{\mathbb{P}}}$, $g\in {\mathcal{F}}_{\mathbb{P},h}$ and $\epsilon >0$,

$$\underset{N\to \infty }{lim\; inf}{m}_N\left(\underset{s\in K}{sup}\left|g\left(s\right)-{\zeta }_{\mathbb{P}}(s+ilh)\right|<\epsilon \right)>0,$$

and the limit

$$\underset{N\to \infty }{lim}{m}_N\left(\underset{s\in K}{sup}\left|g\left(s\right)-{\zeta }_{\mathbb{P}}(s+ilh)\right|<\epsilon \right)$$

exists and is positive for all but at most countably many $\epsilon >0$.

Note that Theorem 4 has a certain advantage over Theorem 3, because it is easier to identify discrete approximating shifts ${\zeta }_{\mathbb{P}}(s+ilh)$ than those continuous ${\zeta }_{\mathbb{P}}(s+i\tau )$. On the other hand, the proof of Theorem 4 is more complicated than that of Theorem 3.

We limit ourselves by the case ${\mathbb{N}}_{\mathbb{P}}\left(x\right)$, only because the case $N_{\mathbb{P}}\left(x^{\alpha }\right)$ with $\alpha \ne 1$ reduces to that $\alpha =1$ after normalization, see [1,15].

We observe that, in place of shifts ${\zeta }_{\mathbb{P}}(s+i\tau )$ and ${\zeta }_{\mathbb{P}}(s+ilh)$, we can consider the approximation by generalized shifts ${\zeta }_{\mathbb{P}}(s+i\phi \left(\tau \right))$ and ${\zeta }_{\mathbb{P}}(s+i\widehat{\phi }\left(l\right))$ with some functions $\phi \left(\tau \right)$ and $\widehat{\phi }\left(l\right)$.

Theorem 4 will be proven in Section 4. Its proof is based on a limit theorem for ${\zeta }_{\mathbb{P}}\left(s\right)$ in the space $\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)$ presented in Section 3. In Section 2, we prepare some mean value estimates and use them for the approximation of ${\zeta }_{\mathbb{P}}\left(s\right)$ in the mean.

It is not easy to present examples of the system $\mathbb{P}$ satisfying (4) and having ${\zeta }_{\mathbb{P}}\left(s\right)$ with a bounded mean square. One example of $\mathbb{P}$ is provided in Conclusions. Searches of such systems $\mathbb{P}$ deserve new papers.

We notice that the most relevant results in the theory of generalized numbers are devoted to the connection of asymmptotics between ${\pi }_{\mathbb{P}}\left(x\right)$ and $N_{\mathbb{P}}\left(x\right)$ as $x\to \infty$, see p. 3.

2. Approximation in the Mean

In this section, we introduce a certain absolutely convergent Dirichlet series, which approximates well in the mean the function ${\zeta }_{\mathbb{P}}\left(s\right)$.

Let $\alpha >1-{\sigma }_{\mathbb{P}}$ be a fixed number, and, for $m\in {\mathbb{N}}_{\mathbb{P}}$ and $n\in \mathbb{N}$, define

$${\kappa }_n\left(m\right)=exp\left\{-{\left({\displaystyle \frac{m}{n}}\right)}^{\alpha }\right\}$$

and

$${\zeta }_{\mathbb{P},n}\left(s\right)=\sum _{m\in {\mathbb{N}}_{\mathbb{P}}}{\displaystyle \frac{{\kappa }_n\left(m\right)}{m^s}}.$$

As ${\kappa }_n\left(m\right){\ll }_n{m}^{-\sigma }$ with every $\sigma$ for every fixed n, the series for ${\zeta }_{\mathbb{P},n}\left(s\right)$ is absolutely convergent, say, for $\sigma >0$.

Define one more function

$$t_n\left(s\right)={\displaystyle \frac{1}{\alpha }}\Gamma \left({\displaystyle \frac{s}{\alpha }}\right)n^s,$$

where $\Gamma \left(s\right)$ is the classical Euler Gamma function. We will use the following integral representation for ${\zeta }_{\mathbb{P},n}\left(s\right)$ from [17].

Lemma 1

([17]). Suppose that the estimate (4) holds. Then, for $s\in D_{{\sigma }_{\mathbb{P}}}$, the integral representation

$${\zeta }_{\mathbb{P},n}\left(s\right)={\displaystyle \frac{1}{2\pi i}}\underset{\alpha -i\infty }{\overset{\alpha +i\infty }{\int }}{\zeta }_{\mathbb{P}}(s+z)t_n\left(z\right)\mathrm{d}z$$

(6)

is valid.

We notice that a proof of Lemma 1 is based on a simple application of the Mellin formula

$${\mathrm{e}}^{-a}=\underset{b-i\infty }{\overset{b+i\infty }{\int }}\Gamma \left(s\right)a^{-s}\mathrm{d}s,a,b>0.$$

The requirement $\sigma >1-{\sigma }_{\mathbb{P}}$ is connected to absolutely convergent Dirichlet series for ${\zeta }_{\mathbb{P}}(s+z)$ in the region $\mathrm{Re}(s+z)>1$.

Now, recall the metric in the space $\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)$, inducing its topology of uniform convergence on compacta. Let $\{K_j:j\in \mathbb{N}\}$ be a sequence of embedded compact sets of the strip $D_{{\sigma }_{\mathbb{P}}}$, such that

$$D_{{\sigma }_{\mathbb{P}}}=\underset{j=1}{\overset{\infty }{\cup }}{K}_j,$$

and each compact subset K of $D_{{\sigma }_{\mathbb{P}}}$ is in some $K_j$. The existence of such a sequence $\left\{K_j\right\}$ is proved in [34] for general regions. Clearly, in the case of strips, we can take a sequence of embedded rectangles with edges parallel to the axis. Then,

$$d(g_1,g_2)=\sum _{j=1}^{\infty }{2}^{-j}{\displaystyle \frac{{sup}_{s\in K_j}|g_1\left(s\right)-g_2\left(s\right)|}{1+{sup}_{s\in K_j}|g_1\left(s\right)-g_2\left(s\right)|}},g_1,g_2\in \mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right),$$

(7)

is a metric of the space $\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)$, which induces the topology of uniform convergence on compact sets.

Before a lemma on the approximation of ${\zeta }_{\mathbb{P}}\left(s\right)$ by ${\zeta }_{\mathbb{P},n}\left(s\right)$, we recall the Gallagher lemma connecting continuous and discrete mean squares of some functions.

Lemma 2

([35]). Suppose that $T_0⩾\delta$, $T⩾\delta$, $\delta >0$, $\mathcal{T}$ is a non-empty finite set in the interval $[T_0+\delta /2,T_0+T-\delta /2]$, and

$$N_{\delta }\left(\tau \right)=\underset{|t-\tau |<\delta }{\sum _{t\in \mathcal{T}}}1,\tau \in \mathcal{T}.$$

Moreover, let the complex-valued function $\mathcal{S}\left(t\right)$ be continuous on $[T_0,T_0+T]$, and have a continuous derivative on $(T_0,T_0+T)$. Then,

$$\sum _{t\in \mathcal{T}}{N}_{\delta }^{-1}{\left(t\right)\left|\mathcal{S}\left(t\right)\right|}^2⩽{\displaystyle \frac{1}{\delta }}\underset{T_0}{\overset{T_0+T}{\int }}{\left|\mathcal{S}\left(t\right)\right|}^2\mathrm{d}t+{\left(\underset{T_0}{\overset{T_0+T}{\int }}{\left|\mathcal{S}\left(t\right)\right|}^2\mathrm{d}t\underset{T_0}{\overset{T_0+T}{\int }}{\left|{\mathcal{S}}^{\prime }\left(t\right)\right|}^2\mathrm{d}t\right)}^{1/2}.$$

Now, we state a result on the approximation of ${\zeta }_{\mathbb{P}}\left(s\right)$ by ${\zeta }_{\mathbb{P},n}\left(s\right)$ in the mean.

Lemma 3.

Suppose that the estimate (4) holds. Then, for every fixed $h>0$,

$$\underset{n\to \infty }{lim}\left(\underset{N\to \infty }{lim\; sup}{\displaystyle \frac{1}{N+1}}\sum _{l=0}^{N}d\left({\zeta }_{\mathbb{P}}(s+ilh),{\zeta }_{\mathbb{P},n}(s+ilh)\right)\right)=0.$$

Proof. 

In view of (7), we have to show that, for every compact set $K\subset D_{{\sigma }_{\mathbb{P}}}$,

$$\underset{n\to \infty }{lim}\left(\underset{N\to \infty }{lim\; sup}{\displaystyle \frac{1}{N+1}}\sum _{l=0}^N\underset{s\in K}{sup}\left|{\zeta }_{\mathbb{P}}(s+ilh)-{\zeta }_{\mathbb{P},n}(s+ilh)\right|\right)=0.$$

(8)

Let us fix a compact set $K\subset D_{{\sigma }_{\mathbb{P}}}$. As K is closed, there is $\epsilon >0$, such that the inequalities ${\sigma }_{\mathbb{P}}+2\epsilon ⩽\sigma ⩽1-\epsilon$ are valid for all $\sigma +it\in K$. Let us take ${\alpha }_1={\sigma }_{\mathbb{P}}+\epsilon -\sigma$. Then ${\alpha }_1<0$, and ${\alpha }_1⩾{\sigma }_{\mathbb{P}}+\epsilon -1+\epsilon ={\sigma }_{\mathbb{P}}+2\epsilon -1>-1$. Let $\alpha =1$. Then we have $0\in ({\alpha }_1,\alpha )$ and $1-\sigma \in ({\alpha }_1,\alpha )$. Therefore, the integrand in (6) possesses only simple poles at $z=0$ and $z=1-s$, the poles of $\Gamma \left(z\right)$ and ${\zeta }_{\mathbb{P}}(s+z)$, respectively, lying in the strip ${\alpha }_1<\mathrm{Re}z<\alpha$. In [15], it was observed that (4) implies the finite order for ${\zeta }_{\mathbb{P}}\left(s\right)$ in the strip to the left of the line $\sigma =1$. Moreover, for $\Gamma \left(s\right)$, the bound

$$\Gamma \left(s\right)\ll exp\{-c|t\left|\right\},$$

(9)

uniform in every strip ${\sigma }_1⩽\sigma ⩽{\sigma }_2$, is valid. Therefore, the representation (6) of Lemma 1 together with the residue theorem provides

$${\zeta }_{\mathbb{P},n}\left(s\right)-{\zeta }_{\mathbb{P}}\left(s\right)={\displaystyle \frac{1}{2\pi i}}\underset{{\alpha }_1-i\infty }{\overset{{\alpha }_1+i\infty }{\int }}{\zeta }_{\mathbb{P}}(s+z)t_n\left(z\right)\mathrm{d}z+\underset{z=1-s}{\mathrm{Res}}{\zeta }_{\mathbb{P}}(s+z)t_n\left(s\right)$$

for all $s\in K$. This and the definition of ${\alpha }_1$, for $s\in K$, implies

$$\begin{array}{ccc}{\zeta }_{\mathbb{P},n}(s+ilh)-{\zeta }_{\mathbb{P}}(s+ilh)\hfill & =\hfill & {\displaystyle \frac{1}{2\pi }}\underset{-\infty }{\overset{\infty }{\int }}{\zeta }_{\mathbb{P}}\left({\sigma }_{\mathbb{P}}+\epsilon +ilh+it+iv\right)t_n\left({\sigma }_{\mathbb{P}}+\epsilon -\sigma +iv\right)\mathrm{d}v\hfill \\ & & +a{t}_n(1-s-ilh)\hfill \\ & =\hfill & {\displaystyle \frac{1}{2\pi }}\underset{-\infty }{\overset{\infty }{\int }}{\zeta }_{\mathbb{P}}\left({\sigma }_{\mathbb{P}}+\epsilon +ilh+iv\right)t_n\left({\sigma }_{\mathbb{P}}+\epsilon -s+iv\right)\mathrm{d}v\hfill \\ & & +a{t}_n(1-s-ilh)\hfill \end{array}$$

after a shift $t+v$ to v. Hence, for $s\in K$,

$$\begin{array}{cc}{\zeta }_{\mathbb{P},n}(s+ilh)-{\zeta }_{\mathbb{P}}(s+ilh)\ll \hfill & \underset{-\infty }{\overset{\infty }{\int }}\left|{\zeta }_{\mathbb{P}}\left({\sigma }_{\mathbb{P}}+\epsilon +ilh+iv\right)\right|\underset{s\in K}{sup}\left|t_n\left({\sigma }_{\mathbb{P}}+\epsilon -s+iv\right)\right|\mathrm{d}v\hfill \\ & +\left|a\right|\underset{s\in K}{sup}\left|t_n(1-s-ilh)\right|,\hfill \end{array}$$

thus

$$\begin{array}{cc}{\displaystyle \frac{1}{N+1}}\sum _{l=0}^N\underset{s\in K}{sup}\hfill & \left|{\zeta }_{\mathbb{P},n}(s+ilh)-{\zeta }_{\mathbb{P}}(s+ilh)\right|\hfill \\ & \ll \underset{-\infty }{\overset{\infty }{\int }}\left({\displaystyle \frac{1}{N+1}}\sum _{l=0}^N\left|{\zeta }_{\mathbb{P}}\left({\sigma }_{\mathbb{P}}+\epsilon +ilh+iv\right)\right|\right)\underset{s\in K}{sup}\left|t_n\left({\sigma }_{\mathbb{P}}+\epsilon -s+iv\right)\right|\mathrm{d}v\hfill \\ & +{\displaystyle \frac{\left|a\right|}{N+1}}\sum _{l=0}^N\underset{s\in K}{sup}\left|t_n(1-s-ilh)\right|\stackrel{\mathrm{def}}{=}I+S.\hfill \end{array}$$

(10)

By the definition of ${\sigma }_{\mathbb{P}}$,

$$\underset{0}{\overset{T}{\int }}{\left|{\zeta }_{\mathbb{P}}\left({\sigma }_{\mathbb{P}}+\epsilon +iv\right)\right|}^{2}dv{\ll }_{\epsilon }T.$$

(11)

From this and the Cauchy integral formula, it follows that

$$\underset{0}{\overset{T}{\int }}{\left|{\zeta }_{\mathbb{P}}^{\prime }\left({\sigma }_{\mathbb{P}}+\epsilon +iv\right)\right|}^{2}dv{\ll }_{\epsilon }T.$$

(12)

For the estimation of the discrete mean of the function ${\zeta }_{\mathbb{P}}\left(s\right)$ in the integral I, we will apply Lemma 2 with $\delta =h$. Thus, we have

$$\begin{array}{cc}\sum _{l=0}^N\hfill & {\left|{\zeta }_{\mathbb{P}}\left({\sigma }_{\mathbb{P}}+\epsilon +ilh+iv\right)\right|}^2{\ll }_h{(1+|v\left|\right)}^A+\underset{h}{\overset{Nh}{\int }}{\left|{\zeta }_{\mathbb{P}}\left({\sigma }_{\mathbb{P}}+\epsilon +iv+i\tau \right)\right|}^2\mathrm{d}\tau \hfill \\ & +{\left(\underset{h}{\overset{Nh}{\int }}{\left|{\zeta }_{\mathbb{P}}\left({\sigma }_{\mathbb{P}}+\epsilon +iv+i\tau \right)\right|}^2\mathrm{d}\tau \underset{h}{\overset{Nh}{\int }}{\left|{\zeta }_{\mathbb{P}}^{\prime }\left({\sigma }_{\mathbb{P}}+\epsilon +iv+i\tau \right)\right|}^2\mathrm{d}\tau \right)}^{1/2}\hfill \\ & {\ll }_{h,\epsilon }{(1+|v\left|\right)}^A+N(t+|v\left|\right),A>0,\hfill \end{array}$$

in view of (11) and (12), and of the finite order for ${\zeta }_{\mathbb{P}}\left(s\right)$. Hence,

$$\begin{array}{cc}{\displaystyle \frac{1}{N+1}}\sum _{l=0}^N\left|{\zeta }_{\mathbb{P}}\left({\sigma }_{\mathbb{P}}+\epsilon +ilh+iv\right)\right|\hfill & {\ll }_{h,\epsilon }{\left({\displaystyle \frac{1}{N+1}}\sum _{l=0}^N{\left|{\zeta }_{\mathbb{P}}\left({\sigma }_{\mathbb{P}}+\epsilon +ilh+iv\right)\right|}^2\right)}^{1/2}\hfill \\ & {\ll }_{h,\epsilon }{(1+|v\left|\right)}^{1/2}+{\displaystyle \frac{{(1+|v\left|\right)}^A}{N+1}}.\hfill \end{array}$$

(13)

From the definition of $t_n\left(s\right)$ and (9), we find, for $s\in K$,

$$\begin{array}{cc}{t}_n\left({\sigma }_{\mathbb{P}}+\epsilon -s+iv\right)\hfill & \ll n^{{\sigma }_{\mathbb{P}}+\epsilon -\sigma }\left|\Gamma \left({\sigma }_{\mathbb{P}}+\epsilon -\sigma +i(v-t)\right)\right|\hfill \\ & \ll n^{-\epsilon }exp\{-c|v-t\left|\right\}{\ll }_K{n}^{-\epsilon }exp\{-c_1\left|v\right|\},c,c_1>0.\hfill \end{array}$$

This and (13) show that

$$I{\ll }_{h,\epsilon ,K}{n}^{-\epsilon }\underset{-\infty }{\overset{\infty }{\int }}\left({(1+|v\left|\right)}^{1/2}+{\displaystyle \frac{1}{N+1}}{(1+|v\left|\right)}^A\right)exp\{-c_1\left|v\right|\}\mathrm{d}v{\ll }_{h,K}{n}^{-\epsilon }$$

(14)

because $\epsilon$ depends on K.

Using similar arguments leads to the following estimate, for $s\in K$,

$$t_n(1-s-ilh)\ll n^{1-\sigma }exp\{-c|lh-t\left|\right\}{\ll }_K{n}^{1-{\sigma }_{\mathbb{P}}-2\epsilon }exp\{-c_{2}lh\},c_2>0.$$

Therefore,

$$S{\ll }_K{\displaystyle \frac{n^{1-{\sigma }_{\mathbb{P}}-2\epsilon }}{N+1}}\sum _{l=0}^{N}exp-c_{2}lh\}{\ll }_K{\displaystyle \frac{n^{1-{\sigma }_{\mathbb{P}}-2\epsilon }}{N+1}}\left(logN+exp\{-c_{2}hlogN\}N\right).$$

Thus, (14) and (10) yield

$${\displaystyle \frac{1}{N+1}}\sum _{l=0}^N\left|{\zeta }_{\mathbb{P}}(s+ilh)-{\zeta }_{\mathbb{P},n}(s+ilh)\right|{\ll }_{h,K}{n}^{-\epsilon }+{\displaystyle \frac{n^{1-{\sigma }_{\mathbb{P}}-2\epsilon }}{N+1}}\left(logN+exp\{-c_{2}hlogN\}N\right).$$

Now, taking $N\to \infty$ and then $n\to \infty$, we obtain (8). The lemma is proven. □

3. Limit Theorem

In this section, we will prove a discrete limit theorem for probability measures in the space $\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)$. Let $\mathcal{X}$ be a certain probability space, $\mathcal{B}\left(\mathcal{X}\right)$ its Borel $\sigma$-field, and Q and $Q_n$, $n\in \mathbb{N}$, probability measures on the measurable space $(\mathcal{X},\mathcal{B}(\mathcal{X}\left)\right)$. From the definition, $Q_n$ converges weakly to Q as $n\to \infty$ ($Q_n\underset{n\to \infty }{\overset{\mathrm{w}}{\to }}Q$) if, for every real continuous bounded function f on $\mathcal{X}$,

$$\underset{n\to \infty }{lim}\underset{\mathcal{X}}{\int }f\mathrm{d}{Q}_n=\underset{\mathcal{X}}{\int }f\mathrm{d}Q.$$

We will consider the weak convergence of the measure

$$P_{\mathbb{P},N,h}\left(A\right)=m_N\left({\zeta }_{\mathbb{P}}(s+ilh)\in A\right),A\in \mathcal{B}\left(\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)\right),$$

as $N\to \infty$. We will prove the following statement.

Proposition 1.

Suppose that estimate (4) holds and $h>0$ is fixed. Then, on $(\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right),\mathcal{B}\left(\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)\right))$, there exists a probability measure $P_{\mathbb{P},h}$ such that $P_{\mathbb{P},N,h}\underset{N\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{\mathbb{P},h}$.

We start the proof of Proposition 1 with a weak convergence on a comparatively simple space $({\Omega }_{\mathbb{P}},\mathcal{B}\left({\Omega }_{\mathbb{P}}\right))$, where

$${\Omega }_{\mathbb{P}}=\prod _{p\in \mathbb{P}}\{z\in \mathbb{C}:|z|=1\}.$$

Thus, ${\Omega }_{\mathbb{P}}$ is an infinite Cartesian product of unit circles on the complex plane. One can equip ${\Omega }_{\mathbb{P}}$ with the product topology and pointwise multiplication operation. Then, ${\Omega }_{\mathbb{P}}$, as a product of compact sets, becomes a compact topological group. For fixed $h>0$ and $A\in \mathcal{B}\left({\Omega }_{\mathbb{P}}\right)$, let

$$P_{N,h}^{{\Omega }_{\mathbb{P}}}\left(A\right)=m_N\left(\left(p^{-ilh}:p\in \mathbb{P}\right)\in A\right).$$

Lemma 4.

For every fixed $h>0$, there exists a probability measure $P_h^{{\Omega }_{\mathbb{P}}}$ on $({\Omega }_{\mathbb{P}},\mathcal{B}\left({\Omega }_{\mathbb{P}}\right))$, such that $P_{N,h}^{{\Omega }_{\mathbb{P}}}\underset{N\to \infty }{\overset{\mathrm{w}}{\to }}{P}_h^{{\Omega }_{\mathbb{P}}}$.

Proof. 

We apply the Fourier transform method, i.e., prove that the Fourier transform of $P_{N,h}^{{\Omega }_{\mathbb{P}}}$ converges, as $N\to \infty$, to a continuous function in discrete topology. Let $\omega =\left(\omega \right(p):p\in \mathbb{P})$ be elements of ${\Omega }_{\mathbb{P}}$. Then, characters of ${\Omega }_{\mathbb{P}}$ are of the form

$$\prod _{p\in \mathbb{P}}{\omega }^{l_p}\left(p\right),$$

where $l_p\in \mathbb{Z}$ and $\#\{l_p:l_p\ne 0\}<\infty$. Hence, the Fourier transform of the measure P on $({\Omega }_{\mathbb{P}},\mathcal{B}\left({\Omega }_{\mathbb{P}}\right))$ is

$$\underset{{\Omega }_{\mathbb{P}}}{\int }\left(\prod _{p\in \mathbb{P}}{\omega }^{l_p}\left(p\right)\right)\mathrm{d}P.$$

(15)

For brevity, let $\mathit{l}=(l_p:l_p\in \mathbb{Z},p\in \mathbb{P})$. Then, in view of (15), the Fourier transform $f_{N,h}^{{\Omega }_{\mathbb{P}}}\left(\mathit{l}\right)$ of the measure $P_{N,h}^{{\Omega }_{\mathbb{P}}}$ of the form

$$\begin{array}{cc}{f}_{N,h}^{{\Omega }_{\mathbb{P}}}\left(\mathit{l}\right)\hfill & =\underset{{\Omega }_{\mathbb{P}}}{\int }\left(\prod _{p\in \mathbb{P}}{\omega }^{l_p}\left(p\right)\right)\mathrm{d}{P}_{N,h}^{{\Omega }_{\mathbb{P}}}={\displaystyle \frac{1}{N+1}}\sum _{l=0}^N\prod _{p\in \mathbb{P}}{p}^{-ilh{l}_p}\hfill \\ & ={\displaystyle \frac{1}{N+1}}\sum _{l=0}^{N}exp\left\{-ilh\sum _{p\in \mathbb{P}}{l}_{p}logp\right\}.\hfill \end{array}$$

Denoting $a\left(\mathit{l}\right)={\sum }_{p\in \mathbb{P}}{l}_{p}logp$, from this, we find

$$f_{N,h}^{{\Omega }_{\mathbb{P}}}\left(\mathit{l}\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}a\left(\mathit{l}\right)={\displaystyle \frac{2\pi r}{h}}\mathrm{with}\mathrm{some}r\in \mathbb{Z},\hfill \\ {\displaystyle \frac{1-exp\{-i(N+1\left)ha\right(\mathit{l}\left)\right\}}{(N+1)(1-exp\{\left\{iha\right(\mathit{l}\left)\right\}\left\}\right)}}\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Thus,

$$\underset{N\to \infty }{lim}{f}_{N,h}^{{\Omega }_{\mathbb{P}}}\left(\mathit{l}\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}a\left(\mathit{l}\right)={\displaystyle \frac{2\pi r}{h}}\mathrm{with}\mathrm{some}r\in \mathbb{Z},\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

This shows that $P_{N,h}^{{\Omega }_{\mathbb{P}}}\underset{N\to \infty }{\overset{\mathrm{w}}{\to }}{P}_h^{{\Omega }_{\mathbb{P}}}$, where the limit measure $P_h^{{\Omega }_{\mathbb{P}}}$ is defined by its Fourier transform

$$f_h^{{\Omega }_{\mathbb{P}}}\left(\mathit{l}\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}a\left(\mathit{l}\right)={\displaystyle \frac{2\pi r}{h}}\mathrm{with}\mathrm{some}r\in \mathbb{Z},\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

□

Now, we return to absolutely convergent Dirichlet series ${\zeta }_{\mathbb{P},n}\left(s\right)$, and consider the weak convergence of

$$P_{\mathbb{P},N,n,h}\left(A\right)\stackrel{\mathrm{def}}{=}{m}_N\left({\zeta }_{\mathbb{P},N,n,h}\in A\right),A\in \mathcal{B}\left(\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)\right),$$

as $N\to \infty$.

Lemma 5.

Suppose that the estimate (4) holds. Thus, for every $h>0$, there is on $(\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right),\mathcal{B}\left(\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)\right))$ a probability measure $P_{\mathbb{P},n,h}$, such that $P_{\mathbb{P},N,n,h}\underset{N\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{\mathbb{P},n,h}$.

Proof. 

The lemma follows easily from Lemma 4 by application the preservation of weak convergence under continuous mappings. First, we continue functions $\left(\omega \right(p)$, $p\in \mathbb{P}$, to the set ${\mathbb{N}}_{\mathbb{P}}$. Let a generalized integer $m\in {\mathbb{N}}_{\mathbb{P}}$ have the representation

$$m=p_1^{{\alpha }_1}\cdots p_r^{{\alpha }_r},p_j\in \mathbb{P},{\alpha }_j\in {\mathbb{N}}_0.$$

We set

$$\omega \left(m\right)={\omega }^{{\alpha }_1}\left(p_1\right)\cdots {\omega }^{{\alpha }_r}\left(p_r\right).$$

(16)

Define a mapping $u_{\mathbb{P},n}:{\Omega }_{\mathbb{P}}\to \mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)$ by

$$u_{\mathbb{P},n}\left(\omega \right)=\sum _{m\in {\mathbb{N}}_{\mathbb{P}}}{\displaystyle \frac{{\kappa }_n\left(m\right)\omega \left(m\right)}{m^s}}.$$

As $\left|\omega \right(m\left)\right|=1$, the latter series is absolutely convergent for $\sigma >0$. Hence, the mapping $u_{\mathbb{P},n}$ is continuous. Moreover, by (16),

$$u_{\mathbb{P},n}\left(p^{-ilh}:p\in \mathbb{P}\right)=\sum _{m\in {\mathbb{N}}_{\mathbb{P}}}{\displaystyle \frac{{\kappa }_n\left(m\right)m^{-ilh}}{m^s}}={\zeta }_{\mathbb{P},n}(s+ilh).$$

Hence, for $A\in \mathcal{B}\left(\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)\right)$,

$$P_{\mathbb{P},N,n,h}\left(A\right)=m_N\left(\left(p^{-ilh}:p\in \mathbb{P}\right)\in u_{\mathbb{P},n}^{-1}A\right),$$

i.e., $P_{\mathbb{P},N,n,h}=P_{N,h}^{{\Omega }_{\mathbb{P}}}{u}_{\mathbb{P},n}^{-1}$, where, for every $A\in \mathcal{B}\left(\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)\right)$,

$$P_{N,h}^{{\Omega }_{\mathbb{P}}}{u}_{\mathbb{P},n}^{-1}\left(A\right)=P_{N,h}^{{\Omega }_{\mathbb{P}}}\left(u_{\mathbb{P},n}^{-1}A\right).$$

This equality, continuity of $u_{\mathbb{P},n}$ and Lemma 4, allow us to apply the principle of preservation of weak convergence under continuous mapping, see Theorem 5.1 of [36]. Thus, setting $P_{\mathbb{P},n,h}\stackrel{\mathrm{def}}{=}{P}_h^{{\Omega }_{\mathbb{P}}}{u}_{\mathbb{P},n}^{-1}$, we obtained the relation $P_{\mathbb{P},N,n,h}\underset{N\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{\mathbb{P},N,h}$. □

For the proof of Proposition 1, we applied Theorem 4.2 from [36] on convergence in the distribution of some random elements. Let $x_n$, $n\in \mathbb{N}$, and x be $\mathcal{X}$-valued random elements defined on a certain probability space $(\widehat{\Omega },\mathcal{B},\nu )$ with distributions $Q_n$ and Q, respectively. Recall that $x_n$ converges to x in the distribution as $n\to \infty$ ($x_n\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}x$), if and only if $Q_n\underset{n\to \infty }{\overset{\mathrm{w}}{\to }}Q$. Lemmas 3 and 5 are used in the application of Theorem 4.2 [36]; however, we also need the weak convergence for $P_{\mathbb{P},n,h}$ as $n\to \infty$. To be precise, the weak convergence for some subsequence $P_{\mathbb{P},n_r,h}$, $n_r\to \infty$ as $r\to \infty$, is sufficient. Thus, we arrive to the notion of relative compactness of the sequence $\{P_{P,n,h}:n\in \mathbb{N}\}$, which means that every subsequence contains a weakly convergent subsequence. The latter property of families of probability measures, in view of the Prokhorov theorem (Theorem 6.1 of [36]), follows from a more convenient tightness property. The tightness of the measure $P_{\mathbb{P},n,h}$ means that for every $\epsilon >0$, there is a compact subset $K=K_{\epsilon }\subset \mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)$, satisfying

$$P_{\mathbb{P},n,h}\left(K\right)>1-\epsilon$$

for all $n\in \mathbb{N}$.

Lemma 6.

Suppose that estimate (4) holds. Then, for all $h>0$, the measure $P_{\mathbb{P},n,h}$ is tight.

Proof. 

Recall that $\left\{K_j\right\}\subset D_{{\sigma }_{\mathbb{P}}}$ is a sequence of compact subsets from the definition of the metric d. We fix a set $K_j$, and take a simple closed contour $L_j$ lying in $D_{{\sigma }_{\mathbb{P}}}$ and enclosing $K_j$. Then, by the Cauchy integral formula, for all $s\in K_j$,

$${\zeta }_{\mathbb{P}}(s+ilh)={\displaystyle \frac{1}{2\pi i}}\underset{L_j}{\int }{\displaystyle \frac{{\zeta }_{\mathbb{P}}(z+ilh)}{s-z}}\mathrm{d}z.$$

This implies

$$\underset{s\in K_j}{sup}\left|{\zeta }_{\mathbb{P}}(s+ilh)\right|\ll \underset{L_j}{\int }{\displaystyle \frac{|{\zeta }_{\mathbb{P}}(z+ilh)\left|\right|\mathrm{d}z|}{|s-z|}}{\ll }_{K_j}{\left(\underset{L_j}{\int }{\left|{\zeta }_{\mathbb{P}}(z+ilh)\right|}^2\left|\mathrm{d}z\right|\right)}^{1/2}.$$

Hence,

$$\begin{array}{cc}{\displaystyle \frac{1}{N+1}}\sum _{l=0}^N\underset{s\in K_j}{sup}\left|{\zeta }_{\mathbb{P}}(s+ilh)\right|\hfill & \ll {\left({\displaystyle \frac{1}{N}}\sum _{l=0}^N\underset{s\in K_j}{sup}{\left|{\zeta }_{\mathbb{P}}(s+ilh)\right|}^2\right)}^{1/2}\hfill \\ & {\ll }_{K_j}{\left(\underset{L_j}{\int }\left({\displaystyle \frac{1}{N}}\sum _{l=0}^N{\left|{\zeta }_{\mathbb{P}}(z+ilh)\right|}^2\right)\left|\mathrm{d}z\right|\right)}^{1/2}{\ll }_{K_j}1<A_j<\infty ,\hfill \end{array}$$

(17)

as an application of Lemma 2 and the bounds

$$\begin{array}{cc}\underset{0}{\overset{T}{\int }}{\left|{\zeta }_{\mathbb{P}}(\mathrm{Rez}+i\mathrm{Imz}+it)\right|}^2\mathrm{d}t\hfill & {\ll }_{K_j}T,\hfill \\ \underset{0}{\overset{T}{\int }}{\left|{\zeta }_{\mathbb{P}}^{\prime }(\mathrm{Rez}+i\mathrm{Imz}+it)\right|}^2\mathrm{d}t\hfill & {\ll }_{K_j}T\hfill \end{array}$$

for $z\in L_j$ give

$${\displaystyle \frac{1}{N}}\sum _{l=0}^N{\left|{\zeta }_{\mathbb{P}}(z+ilh)\right|}^2{\ll }_{K_j}1.$$

Using (8) and (17), we obtain

$$\begin{array}{cc}\underset{n\in \mathbb{N}}{sup}\underset{N\to \infty }{lim\; sup}\hfill & {\displaystyle \frac{1}{N+1}}\sum _{l=0}^N\underset{s\in K_j}{sup}\left|{\zeta }_{\mathbb{P},n}(s+ilh)\right|\hfill \\ & \ll \underset{n\in \mathbb{N}}{sup}\underset{N\to \infty }{lim\; sup}{\displaystyle \frac{1}{N+1}}\sum _{l=0}^N\underset{s\in K_j}{sup}\left|{\zeta }_{\mathbb{P}}(s+ilh)-{\zeta }_{\mathbb{P},n}(s+ilh)\right|\hfill \\ & +\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N+1}}\sum _{l=0}^N\underset{s\in K_j}{sup}\left|{\zeta }_{\mathbb{P}}(s+ilh)\right|\ll {\widehat{A}}_j<\infty .\hfill \end{array}$$

(18)

Now, introduce a random variable ${\xi }_N$ defined on a certain probability space $(\widehat{\Omega },\mathcal{B},\nu )$ and with a distribution

$$\nu \{{\xi }_N=lh\}={\displaystyle \frac{1}{N+1}},l=0,1,\dots ,N,$$

and, based on the above probability space, define the $\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)$-valued random element

$${\zeta }_{\mathbb{P},N,n,h}={\zeta }_{\mathbb{P},N,n,h}\left(s\right)={\zeta }_{\mathbb{P},n}(s+i{\xi }_{N}h),$$

as well as the $\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)$-valued random element ${\zeta }_{\mathbb{P},n,h}={\zeta }_{\mathbb{P},n}\left(s\right)$ with the distribution $P_{\mathbb{P},n,h}$. For a fixed $\epsilon >0$, define $C_j=2^{-j}{\widehat{A}}_j{\epsilon }^{-1}$. Then, inequality (18) together with Lemma 5 yields

$$\begin{array}{cc}\nu \left\{\underset{s\in K_j}{sup}\left|{\zeta }_{\mathbb{P},n,h}\left(s\right)\right|⩾C_j\right\}\hfill & ⩽\underset{n\in \mathbb{N}}{sup}\underset{N\to \infty }{lim\; sup}\nu \left\{\underset{s\in K_j}{sup}\left|{\zeta }_{\mathbb{P},N,n,h}\left(s\right)\right|⩾C_j\right\}\hfill \\ & ⩽\underset{n\in \mathbb{N}}{sup}\underset{N\to \infty }{lim\; sup}{\displaystyle \frac{1}{C_j(N+1)}}\sum _{l=0}^N\underset{s\in K_j}{sup}\left|{\zeta }_{\mathbb{P},N,h}(s+ilh)\right|⩽2^{-j}\epsilon \hfill \end{array}$$

(19)

for all $n\in {\mathbb{N}}_0$. Set

$$K=\left\{g\left(s\right)\in \mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right):\underset{s\in K_j}{sup}\left|g\left(s\right)\right|⩽C_j,j\in \mathbb{N}\right\}.$$

As uniformly bounded on compact sets, the set K is compact in $\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)$. Moreover, in view of (19),

$$\begin{array}{cc}{P}_{\mathbb{P},n,h}\left(K\right)\hfill & =1-P_{\mathbb{P},n,h}(\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)\setminus K)\hfill \\ & =1-P_{\mathbb{P},n,h}\left(g\left(s\right)\in \mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right):\underset{s\in K_j}{sup}\left|{\zeta }_{\mathbb{P},n,h}\left(s\right)\right|⩾C_j\mathrm{for}\mathrm{some}j\right)\hfill \\ & =1-P_{\mathbb{P},n,h}\left(\underset{j=1}{\overset{\infty }{\cup }}\left\{g\left(s\right)\in \mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right):\underset{s\in K_j}{sup}\left|g\left(s\right)\right|⩾C_j\right\}\right)\hfill \\ & =1-\sum _{j=1}^{\infty }{P}_{\mathbb{P},n,h}\left(g\left(s\right)\in \mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right):\underset{s\in K_j}{sup}\left|g\left(s\right)\right|⩾C_j\right)\hfill \\ & =1-\sum _{j=1}^{\infty }\nu \left\{\underset{s\in K_j}{sup}\left|{\zeta }_{\mathbb{P},n,h}\left(s\right)\right|⩾C_j\right\}=1-\epsilon \sum _{j=1}^{\infty }{2}^{-j}=1-\epsilon \hfill \end{array}$$

for all $n\in \mathbb{N}$. The proof of the lemma is complete. □

Proof of Proposition 1. 

We will show that hypotheses of Theorem 4.2 from [36] are fulfilled in our case. We recall this theorem. Suppose that the space $(\mathcal{X},d)$ is separable, $x_{nl}$ and $y_n$ are $\mathcal{X}$-valued random elements defined on the probability space $(\widehat{\Omega },\mathcal{B},\nu )$, $x_{nl}\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}{x}_l$, $x_l\underset{l\to \infty }{\overset{\mathcal{D}}{\to }}x$, and, for every $\epsilon >0$,

$$\underset{l\to \infty }{lim}\underset{n\to \infty }{lim\; sup}\nu \left(d(y_n,x_{nl})⩾\epsilon \right)=0.$$

Then, $y_n\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}x$.

Lemma 6 and Prokhorov’s theorem (Theorem 6.1 of [36]) imply that the sequence $\{P_{\mathbb{P},n,h}:n\in \mathbb{N}\}$ is relatively compact. Hence, there exists a subsequence $\left\{P_{\mathbb{P},n_l,h}\right\}\subset P_{\mathbb{P},n,h}$ and a probability measure $P_{\mathbb{P},h}$ on $(\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right),\mathcal{B}\left(\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)\right))$, such that $P_{\mathbb{P},n_l,h}\underset{l\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{\mathbb{P},h}$. This relation can be written in the form

$${\zeta }_{\mathbb{P},n_l,h}\underset{l\to \infty }{\overset{\mathcal{D}}{\to }}{P}_{\mathbb{P},h}.$$

(20)

The assertion of Lemma 5 is equivalent to

$${\zeta }_{\mathbb{P},N,n_l,h}\underset{N\to \infty }{\overset{\mathcal{D}}{\to }}{\zeta }_{\mathbb{P},n_l,h}.$$

(21)

Introduce one more $\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)$-valued random element

$${\zeta }_{\mathbb{P},N,h}={\zeta }_{\mathbb{P},N,h}\left(s\right)=\zeta (s+i{\xi }_{N}h).$$

Then, by Lemma 3, for every $\epsilon >0$, we have

$$\begin{array}{cc}\underset{l\to \infty }{lim}\underset{N\to \infty }{lim\; sup}\hfill & \nu \left\{d\left({\zeta }_{\mathbb{P},N,n_l,h},{\zeta }_{\mathbb{P},N,h}\right)⩾\epsilon \right\}\hfill \\ & =\underset{l\to \infty }{lim}\underset{N\to \infty }{lim\; sup}{m}_N\left(d\left({\zeta }_{\mathbb{P}}(s+ilh),{\zeta }_{\mathbb{P},n_l}(s+ilh)\right)⩾\epsilon \right)\hfill \\ & ⩽\underset{l\to \infty }{lim}\underset{N\to \infty }{lim\; sup}{\displaystyle \frac{1}{(N+1)\epsilon }}\sum _{l=0}^{N}d\left({\zeta }_{\mathbb{P}}(s+ilh),{\zeta }_{\mathbb{P},n_l}(s+ilh)\right)=0.\hfill \end{array}$$

This, and relations (20) and (21) ensure an application of Theorem 4.2 of [36]. Thus, ${\zeta }_{\mathbb{P},N,h}\underset{N\to \infty }{\overset{\mathcal{D}}{\to }}{P}_{\mathbb{P},h}$ gives

$$P_{\mathbb{P},N,h}\underset{N\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{\mathbb{P},h}.$$

□

4. Proof of Approximation

In this section, using Proposition 1, we will prove Theorem 4. The main role in the proof belongs to the measure $P_{\mathbb{P},h}$ and its support. Recall that the support of the measure $P_{\mathbb{P},h}$ is a minimal closed set $S_{\mathbb{P},h}\subset \mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)$, such that $P_{\mathbb{P},h}\left(S_{\mathbb{P},h}\right)=1$. We observe that

$$S_{\mathbb{P},h}=\{g\in \mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right):P_{\mathbb{P},h}\left(G\right)>0\mathrm{for}\mathrm{every}\mathrm{open}\mathrm{neighbourhood}G\mathrm{of}g\}.$$

(22)

Also, we will use two equivalents of weak convergence of probability measures.

Lemma 7.

Let $Q_n$, $n\in \mathbb{N}$, and Q be probability measures on $(\mathcal{X},\mathcal{B}(\mathcal{X}\left)\right)$. Then, the following statements are equivalent:

(i) 

$Q_n\underset{n\to \infty }{\overset{\mathrm{w}}{\to }}Q$;

(ii) 

For every open set $G\subset \mathcal{X}$,

$$\underset{n\to \infty }{lim\; inf}{Q}_n\left(G\right)⩾Q\left(G\right);$$

(iii) 

For every continuity set A of the measure Q ($Q(\partial A)=0$, where $\partial A$ denotes the boundary of the set A),

$$\underset{n\to \infty }{lim}{Q}_n\left(A\right)=Q\left(A\right).$$

The lemma is a part of Theorem 2.1 from [36], where the proof is given.

Proof of Theorem 4. 

Suppose that ${\mathcal{F}}_{\mathbb{P},h}$ is the support of the limit measure $P_{\mathbb{P},h}$ in Proposition 1, i.e., ${\mathcal{F}}_{\mathbb{P},h}=S_{\mathbb{P},h}$. Then, from the definition of the support, ${\mathcal{F}}_{\mathbb{P},h}$ is a closed set, and $P_{\mathbb{P},h}\left({\mathcal{F}}_{\mathbb{P},h}\right)=1$, hence ${\mathcal{F}}_{\mathbb{P},h}\ne ⌀$.

For $g\in {\mathcal{F}}_{\mathbb{P},h}$, a compact set $K\subset D_{{\sigma }_{\mathbb{P}}}$ and $\epsilon >0$, define

$$G_{\mathbb{P},\epsilon }=\left\{f\in \mathcal{H}\left(S_{{\sigma }_{\mathbb{P}}}\right):\underset{s\in K}{sup}|f\left(s\right)-g\left(s\right)|<\epsilon \right\}.$$

As the set K is compact, $G_{\mathbb{P},\epsilon }$ is an open set in $\mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right)$. Therefore, the application of Proposition 1, and (i) and (ii) of Lemma 7, yields

$$\underset{N\to \infty }{lim\; inf}{P}_{P,N,h}\left(G_{\mathbb{P},\epsilon }\right)⩾P_{\mathbb{P},h}\left(G_{\mathbb{P},\epsilon }\right).$$

(23)

The set $G_{\mathbb{P},\epsilon }$ is an open neighbourhood of an element g of the support of the measure $P_{\mathbb{P},h}$. Therefore, in view of (22),

$$P_{\mathbb{P},h}\left(G_{\mathbb{P},\epsilon }\right)>0.$$

(24)

The definitions of the measure $P_{\mathbb{P},N,h}$ and the set $G_{\mathbb{P},\epsilon }$ together with inequalities (23) and (24) provide the first statement

$$\underset{N\to \infty }{lim\; inf}{m}_N\left(\underset{s\in K}{sup}\left|g\left(s\right)-{\zeta }_{\mathbb{P}}(s+ilh)\right|<\epsilon \right)>0$$

of Theorem 4.

In order to prove the second statement of Theorem 4, we apply (i) and (iii) of Lemma 7. We observe that the boundary $\partial G_{\mathbb{P},ve}$ lies in the set

$$\left\{f\in \mathcal{H}\left(D_{{\sigma }_{\mathbb{P}}}\right):\underset{s\in K}{sup}|f\left(s\right)-g\left(s\right)|=\epsilon \right\}.$$

From this, we see that $\partial G_{\mathbb{P},{\epsilon }_1}\ne \partial G_{\mathbb{P},{\epsilon }_2}$ if ${\epsilon }_1\ne {\epsilon }_2$. This remark implies that $P_{\mathbb{P},h}(\partial G_{\mathbb{P},\epsilon })>0$ for, at most, countably many values of $\epsilon >0$. Therefore, Proposition 1, and (i) and (iii) of Lemma 7, imply the equality

$$\underset{N\to \infty }{lim}{P}_{\mathbb{P},N,h}\left(G_{\mathbb{P},\epsilon }\right)=P_{\mathbb{P},h}\left(G_{\mathbb{P},\epsilon }\right)$$

for all but, at most, countably many $\epsilon >0$. The latter equality, inequality (24), and the definitions of $P_{\mathbb{P},N,h}$ and $G_{\mathbb{P}\epsilon }$ show that the limit

$$\underset{N\to \infty }{lim}{m}_N\left(\underset{s\in K}{sup}\left|g\left(s\right)-{\zeta }_{\mathbb{P}}(s+ilh)\right|<\epsilon \right)$$

exists and is positive for all but, at most, countably many $\epsilon >0$. The theorem is proved. □

5. Conclusions

In the present paper, we found that the Beurling zeta function ${\zeta }_{\mathbb{P}}\left(s\right)$, $s=\sigma +it$,

$${\zeta }_{\mathbb{P}}\left(s\right)=\prod _{p\in \mathbb{P}}{\left(1-{\displaystyle \frac{1}{p^s}}\right)}^{-1}{,}^{\prime }\sigma >{\sigma }_{\mathbb{P}},$$

of the system $\mathbb{P}$ of generalized prime numbers whose system ${\mathbb{N}}_{\mathbb{P}}$ of generalized integers satisfies the estimate

$$\underset{m\in {\mathbb{N}}_{\mathbb{P}}}{\sum _{m⩽x}}1-ax{\ll }_{\mathbb{P}}{x}^{\delta },0⩽\delta <1,$$

(25)

and having a bounded mean square

$$\underset{0}{\overset{T}{\int }}{\left|{\zeta }_{\mathbb{P}}(\sigma +it)\right|}^2\ll T,\delta <\sigma <1,$$

(26)

has good discret-type approximation properties. This means that the discrete shifts ${\zeta }_{\mathbb{P}}(s+ilh)$, $h>0$, $l\in {\mathbb{N}}_0$, approximate all analytic functions of a certain class.

We will provide an example of the system $\mathbb{P}$. Denote by q the rational prime numbers, and set

$$\mathbb{P}=\left\{2,q\equiv 1\left(\mathrm{mod}4\right)\mathrm{with}\mathrm{multiplicity}2,\sqrt{q}:q\equiv 3\left(\mathrm{mod}4\right)\right\}.$$

It is known [37] that (25) is true with $\delta =23/73$. The latter system was discussed in [15]. Moreover, from [38], it follows that (26) holds for $\sigma >(1+\delta )/2=48/73$. Therefore, in the case of the example, the approximated analytic functions are defined in the strip $D_{{\sigma }_{\mathbb{P}}}=\{s\in \mathbb{C}:48/73<\sigma <1\}$.

We are planning to extend Theorem 4 for generalized shifts ${\zeta }_{\mathbb{P}}(s+i\phi \left(l\right))$ with a certain function $\phi \left(l\right)$. This also concerns the case of continuous shifts ${\zeta }_{\mathbb{P}}(s+i\phi \left(\tau \right))$.