On Value Distribution of Certain Beurling Zeta-Functions

Abstract

In this paper, the approximation of analytic functions by shifts ${\zeta }_{\mathcal{P}}(s+i\tau )$ of Beurling zeta-functions ${\zeta }_{\mathcal{P}}\left(s\right)$ of certain systems $\mathcal{P}$ of generalized prime numbers is discussed. It is required that the system of generalized integers ${\mathcal{N}}_{\mathcal{P}}$ generated by $\mathcal{P}$ satisfies ${\sum }_{m⩽x,m\in \mathcal{N}}1=ax+O\left(x^{\delta }\right)$, $a>0$, $0⩽\delta <1$, and the function ${\zeta }_{\mathcal{P}}\left(s\right)$ in some strip lying in $\widehat{\sigma }<\sigma <1$, $\widehat{\sigma }>\delta$, which has a bounded mean square. Proofs are based on the convergence of probability measures in some spaces.

1. Introduction

The Riemann zeta-function $\zeta \left(s\right)$, $s=\sigma +it$, is defined, for $\sigma >1$ by

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

where the product is taken over prime numbers q, has a meromorphic continuation to the complex plane with the unique simple pole $s=1$, ${\mathrm{Re}}_{s=1}\zeta \left(s\right)=1$ (see, for example, [1]), and has several generalizations. One of them is Beurling zeta-functions.

The system $\mathcal{P}$ of real numbers $1<p_1⩽p_2⩽\cdots ⩽p_n⩽\cdots$, $p_n\to \infty$ as $n\to \infty$, is called generalized prime numbers. From numbers of system $\mathcal{P}$, the system ${\mathcal{N}}_{\mathcal{P}}$ of generalized integers

$$p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\cdots p_r^{{\alpha }_r}\cdots ,{\alpha }_j\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\},j=1,\dots ,r,\dots ,$$

is obtained. As in the theory of rational primes q, the main attention is devoted to asymptotics of the function

$${\pi }_{\mathcal{P}}\left(x\right)=\underset{p\in \mathcal{P}}{\sum _{p⩽x}}1,x\to \infty .$$

Together with ${\pi }_{\mathcal{P}}\left(x\right)$, the number of generalized integers m

$${\mathcal{N}}_{\mathcal{P}}\left(x\right)=\underset{m\in {\mathcal{N}}_{\mathcal{P}}}{\sum _{m⩽x}}1,x\to \infty ,$$

is considered. The above sums are taken by counting multiplicities of p and m, respectively. By the Landau result [2], it is known that the estimate

$${\mathcal{N}}_{\mathcal{P}}\left(x\right)=ax+O\left(x^{\delta }\right),0⩽\delta <1,a>0,$$

(1)

implies

$${\pi }_{\mathcal{P}}\left(x\right)={\displaystyle \underset{2}{\overset{x}{\int }}}\frac{\mathrm{d}u}{logu}+O\left(x{\mathrm{e}}^{-c\sqrt{logx}}\right),c>0.$$

The distribution of generalized numbers was studied by Beurling [3], Borel [4], Diamond [5,6,7], Mallavin [8], Nyman [9], Ryavec [10], Stankus [11], Zhang [12], Hilberdink and Lapidus [13], Schlage-Puhta and Vindas [14], Debruyne, Schlage-Puhta and Vindas [15], and others. Among other problems studied in the above works, the central place is occupied by the relation between

$${\mathcal{N}}_{\mathcal{P}}\left(x\right)=ax+O\left(\frac{x}{{(\log x)}^{\alpha }}\right),\alpha >0,$$

(2)

and

$${\pi }_{\mathcal{P}}\left(x\right)={\displaystyle \underset{2}{\overset{x}{\int }}}\frac{\mathrm{d}u}{\log u}+O\left(\frac{x}{{(\log x)}^{\beta }}\right),\beta >0.$$

For example, in [9], it was obtained that the above estimates with arbitrary $\alpha$ and $\beta$ are equivalent. The papers [6,8,16] are devoted to formulae for ${\pi }_{\mathcal{P}}\left(x\right)$, with the remainder term of order $O\left(x{\mathrm{e}}^{-c_1{(logx)}^{\beta }}\right)$ implied by ${\mathcal{N}}_{\mathcal{P}}\left(x\right)$ with the remainder term $O\left(x{\mathrm{e}}^{-c_2{(logx)}^{\alpha }}\right)$. Beurling proved [3] that the asymptotics

$${\pi }_{\mathcal{P}}\left(x\right)\sim \frac{x}{logx},x\to \infty ,$$

(3)

follows from (2) with $\alpha >3/2$, and this is not true with $\alpha =3/2$ for all systems of generalized primes. Moreover, for the investigation of ${\pi }_{\mathcal{P}}\left(x\right)$, he introduced the zeta-functions ${\zeta }_{\mathcal{P}}\left(s\right)$ defined in some half-planes by the Euler product

$${\zeta }_{\mathcal{P}}\left(s\right)=\prod _{p\in \mathcal{P}}{\left(1-\frac{1}{p^s}\right)}^{-1},$$

or by the Dirichlet series

$${\zeta }_{\mathcal{P}}\left(s\right)={\displaystyle \sum _{m\in {\mathcal{N}}_{\mathcal{P}}}^{\infty }}\frac{1}{m^s}.$$

The convergence of the latter objects depends on the system $\mathcal{P}$ of generalized primes.

It is easily seen that in case (1), the series for ${\zeta }_{\mathcal{P}}\left(s\right)$ is absolutely convergent for $\sigma >1$. Actually, the partial summation formula shows that

$$\underset{m\in {\mathcal{N}}_{\mathcal{P}}}{\sum _{m⩽x}}\frac{1}{m^s}=\frac{1}{x^s}{\mathcal{N}}_{\mathcal{P}}\left(x\right)+s{\displaystyle \underset{1}{\overset{x}{\int }}}\frac{{\mathcal{N}}_{\mathcal{P}}\left(x\right)}{x^{s+1}}\mathrm{d}x.$$

(4)

Since, for $\sigma >1$, the integral

$${\displaystyle \underset{1}{\overset{\infty }{\int }}}\frac{{\mathcal{N}}_{\mathcal{P}}\left(x\right)}{x^{s+1}}\mathrm{d}x$$

is absolutely and uniformly convergent for $\sigma ⩾1+\epsilon$, $\forall \epsilon >0$, and $x^{-s}{\mathcal{N}}_{\mathcal{P}}\left(x\right)=o\left(1\right)$, so from (4) we have

$${\zeta }_{\mathcal{P}}\left(s\right)=s{\displaystyle \underset{1}{\overset{\infty }{\int }}}\frac{{\mathcal{N}}_{\mathcal{P}}\left(x\right)}{x^{s+1}}\mathrm{d}x.$$

(5)

Thus, ${\zeta }_{\mathcal{P}}\left(s\right)$ is analytic in the half-plane $\sigma >1$. Moreover, in this half-plane,

$$\prod _{p\in \mathcal{P}}{\left(1-\frac{1}{p^s}\right)}^{-1}=\sum _{m\in {\mathcal{N}}_{\mathcal{P}}}\frac{1}{m^s}.$$

Now, the functions ${\zeta }_{\mathcal{P}}\left(s\right)$ are called Beurling zeta-functions.

As it was observed by Beurling [3], it suffices to consider ${\mathcal{N}}_{\mathcal{P}}\left(x\right)$ in place of ${\mathcal{N}}_{\mathcal{P}}\left(x^{\delta }\right)$, $\delta \ne 1$, because the latter case reduces after normalization to ${\mathcal{N}}_{\mathcal{P}}\left(x\right)$.

An important problem is the analytic continuation of the function ${\zeta }_{\mathcal{P}}\left(s\right)$. Suppose that (1) is true. Then, (5) implies

$${\zeta }_{\mathcal{P}}\left(s\right)=\frac{as}{s-1}+s{\displaystyle \underset{1}{\overset{\infty }{\int }}}\frac{r\left(x\right)}{x^{s+1}}\mathrm{d}x,r\left(x\right)=O\left(x^{\delta }\right),\delta <1,$$

the latter integral being absolutely and uniformly convergent for $\sigma ⩾\delta +\epsilon$, $\forall \epsilon >0$. Therefore, the function ${\zeta }_{\mathcal{P}}\left(s\right)$ has analytic continuation to the half-plane $\sigma >\delta$, except for a simple pole at the point $s=1$ with residue a.

Much attention is devoted to analytic continuation for the function ${\zeta }_{\mathcal{P}}\left(s\right)$ in [13]. For this, the generalized von Mongoldt function

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

and

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

are used. Let

$${\psi }_{\mathcal{P}}\left(x\right)=x+O\left(x^{\alpha +\epsilon }\right),\alpha \in [0,1),\forall \epsilon >0.$$

Then, in [13], it is proved that ${\zeta }_{\mathcal{P}}\left(s\right)$ has an analytic continuation to the half-plane $\sigma >\alpha$, except for a simple pole at the point $s=1$. Under certain additional conditions, the latter estimate is necessary as well.

There is another method for the analytic continuation of ${\zeta }_{\mathcal{P}}\left(s\right)$ cultivated in [13]. However, for our aims, we limit ourselves by the analytic continuation to the half-plane $\sigma >\delta$ because, throughout the paper, we suppose the validity of the axiom (1).

The paper [17] is devoted to zero-distribution of ${\zeta }_{\mathcal{P}}\left(s\right)$, where various zero-density results corresponding to those of $\zeta \left(s\right)$ are given. We stress that in [17], the Beurling prime number theorem [3] was strengthened, and it was proved that asymptotics (3) is implied by the estimate of Cesàro type

$${\displaystyle \underset{1}{\overset{x}{\int }}}\frac{{\mathcal{N}}_{\mathcal{P}}\left(t\right)-at}{t}{\left(1-\frac{t}{x}\right)}^m\mathrm{d}t=O\left(\frac{x}{{(logx)}^{\alpha }}\right),\alpha >\frac{3}{2},x\to \infty ,$$

with some $m\in \mathbb{N}$.

In the present paper, differently from the cited above works, including [14,17], that are devoted to prime number theorem, analytic continuation and zeros of ${\zeta }_{\mathcal{P}}\left(s\right)$, we focus on the approximation properties of the Beurling zeta-functions. More precisely, we consider the approximation of a set of analytic functions $f\left(s\right)$ by shifts ${\zeta }_{\mathcal{P}}(s+i\tau )$, $\tau \in \mathbb{R}$, i.e., such $\tau$ that, for some compact sets K and $\epsilon >0$,

$$\underset{s\in K}{sup}\left|{\zeta }_{\mathcal{P}}(s+i\tau )-f\left(s\right)\right|<\epsilon .$$

The case of the Riemann zeta-function shows that the results of such a type have serious theoretical (functional independence, zero-distribution, moment problem, …) and practical (approximation theory, quantum mechanics) applications, see [18]. Moreover, investigations of the approximation of analytic functions by zeta-functions have an impact on the Linnik–Ibragimov conjecture on the universality of the Dirichlet series; see Section 1.6 of [19].

For our aims, the mean square estimate for ${\zeta }_{\mathcal{P}}\left(s\right)$ is needed. Let

$$M(\sigma ,T)\stackrel{\mathrm{def}}{=}{\displaystyle \underset{0}{\overset{T}{\int }}}{\left|{\zeta }_{\mathcal{P}}(\sigma +it)\right|}^2\mathrm{d}t,$$

and $\widehat{\sigma }=inf\{\sigma :M(\sigma ,T){\ll }_{\sigma }T,\sigma >\delta \}$. Suppose that $\widehat{\sigma }<1$, and define $D_{\mathcal{P}}=\{s\in \mathbb{C}:\widehat{\sigma }<\sigma <1\}$. Here, and in what follows, the notation $z{\ll }_{\epsilon }y$, $z\in \mathbb{C}$, $y>0$ is a synonym of $z=O\left(y\right)$ with implied constant depending on $\epsilon$. Denote by $\mathcal{H}\left(D_{\mathcal{P}}\right)$ the space of analytic on $D_{\mathcal{P}}$ functions endowed with the topology of uniform convergence on compacta.

It is well-known that the Riemann zeta-function $\zeta \left(s\right)$ and some other zeta-functions are universal, i.e., their shifts $\zeta (s+i\tau )$, $\tau \in \mathbb{R}$ are approximately defined in certain strip analytic functions; see [18,19,20,21,22,23,24,25] for results and problems. We believe that the function ${\zeta }_{\mathcal{P}}\left(s\right)$ for some systems of generalized prime numbers $\mathcal{P}$ also has similar approximation properties. However, every case of system $\mathcal{P}$ requires a separate investigation. In the paper, we propose the following result for the approximation of analytic functions by shifts ${\zeta }_{\mathcal{P}}(s+i\tau )$. In what follows, $m_{L}A$ denotes the Lebesgue measure of $A\subset \mathbb{R}$. The main result of the paper is the following theorem.

Theorem 1.

Assume that the system $\mathcal{P}$ satisfies the axiom (1). Then, there exists a non-empty closed subset $F_{\mathcal{P}}\subset \mathcal{H}\left(D_{\mathcal{P}}\right)$, such that, for all compact sets $K\subset D_{\mathcal{P}}$, $f\left(s\right)\in F_{\mathcal{P}}$ and $\epsilon >0$,

$$\underset{T\to \infty }{lim\; inf}\frac{1}{T}{m}_L\left\{\tau \in [0,T]:\underset{s\in K}{sup}\left|{\zeta }_{\mathcal{P}}(s+i\tau )-f\left(s\right)\right|<\epsilon \right\}>0.$$

In addition, the limit

$$\underset{T\to \infty }{lim}\frac{1}{T}{m}_L\left\{\tau \in [0,T]:\underset{s\in K}{sup}\left|{\zeta }_{\mathcal{P}}(s+i\tau )-f\left(s\right)\right|<\epsilon \right\}$$

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

Theorem 1 will be proved in Section 5.

Let $\mathcal{B}\left(\mathbb{X}\right)$ stand for the Borelean $\sigma$-field of the topological space $\mathbb{X}$, and, for $A\in \mathcal{B}\left(\mathcal{H}\left(D_{\mathcal{P}}\right)\right)$,

$$P_{T,\mathcal{P}}\left(A\right)=\frac{1}{T}{m}_L\left\{\tau \in [0,T]:{\zeta }_{\mathcal{P}}(s+i\tau )\in A\right\}.$$

Theorem 1 will be derived from the next theorem on weak convergence of $P_{T,\mathcal{P}}$ as $T\to \infty$.

Theorem 2.

Suppose that the system $\mathcal{P}$ satisfies the axiom (1). Then $P_{T,\mathcal{P}}$, as $T\to \infty$, weakly converges to a certain measure $P_{\mathcal{P}}$ on $(\mathcal{H}\left(D_{\mathcal{P}}\right),\mathcal{B}\left(\mathcal{H}\left(D_{\mathcal{P}}\right)\right))$.

Theorem 2 will be proved in Section 4.

We recall some examples connected to the hypotheses of Theorems 1 and 2.

A problem of the validity of axiom (1) is not easy. The following interesting example is known; see [13]. Let the system of generalized integers ${\mathcal{N}}_{\mathcal{P}}$ be generated by the system

$$\mathcal{P}=(2,\sqrt{3},5,5,\sqrt{7},\sqrt{11},13,13,\dots ),$$

i.e., $\mathcal{P}$ includes 2, rational primes $q\equiv 1\left(\mathrm{mod}4\right)$ with multiplicity 2, and $\sqrt{q}$ with rational primes $q\equiv 3\left(\mathrm{mod}4\right)$. Then, it is known that

$${\mathcal{N}}_{\mathcal{P}}\left(x\right)=\frac{\pi }{4}x+O\left(x^{23/73}\right).$$

In [11], the system $\mathcal{P}$ of shifted rational primes $q=\pi \left(r\right)+1$ with $r>0$, $\pi \left(r\right)={\sum }_{q⩽r}1$, was considered, and it was obtained that

$${\mathcal{N}}_{\mathcal{P}}\left(x\right)=ax+O\left(xexp\left\{-\left(1-\frac{c{log}_{3}x}{{log}_{2}x}\right)\sqrt{\frac{1}{2}logx{log}_{2}x}\right\}\right),$$

where ${log}_{n}x={\underbrace{log\dots log}}_{n}x$, $a>0$, $c>0$. This shows that the estimate (1), even for a comparatively simple system $\mathcal{P}$, is difficult to reach.

Write generalized numbers in another form

$$1={\nu }_1<{\nu }_2<\cdots$$

with corresponding multiplicities $1=a_1,a_2,\dots$. Then, we have

$${\mathcal{N}}_{\mathcal{P}}\left(x\right)=\sum _{{\nu }_m⩽x}{a}_m,$$

and

$${\zeta }_{\mathcal{P}}\left(s\right)={\displaystyle \sum _{m=1}^{\infty }}\frac{a_m}{{\nu }_m^s}.$$

In [26], the following result has been obtained. Suppose that (1) is true, and ${\nu }_{m+1}-{\nu }_m\gg exp\{-{\nu }_m^{\kappa }\}$ with every $\kappa >0$. Then, for $\sigma >(1+\delta )/2$,

$$\underset{T\to \infty }{lim}\frac{1}{2T}{\displaystyle \underset{-T}{\overset{T}{\int }}}{\left|{\zeta }_{\mathcal{P}}(\sigma +it)\right|}^2\mathrm{d}t={\displaystyle \sum _{m=1}^{\infty }}\frac{a_m^2}{{\nu }_m^{2\sigma }}.$$

This implies that $\widehat{\sigma }=(1+\delta )/2<1$ in this case.

We divide the proof of Theorem 2 into parts. We start with weak convergence of probability measures in comparatively simple spaces and finish in the space $\mathcal{H}\left(D_{\mathcal{P}}\right)$.

2. Case of Compact Group

Define the set

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

The elements of $\Omega$ are all functions $\omega :\mathcal{P}\to \{s\in \mathbb{C}:|s|=1\}$. We equipped $\Omega$ with the product topology and operation of pointwise multiplication. Since the unit circle is a compact set, by the Tikhonov theorem [27], $\Omega$ is a compact topological group. For $A\in \mathcal{B}(\Omega )$, set

$$P_{T,\mathcal{P}}^{\Omega }\left(A\right)=\frac{1}{T}{m}_L\left\{\tau \in [0,T]:\left(p^{-i\tau }:p\in \mathcal{P}\right)\in A\right\}.$$

Lemma 1.

$P_{T,\mathcal{P}}^{\Omega }$ weakly converges to a certain measure $P_{\mathcal{P}}^{\Omega }$ on $(\Omega ,\mathcal{B}(\Omega \left)\right)$ as $T\to \infty$.

Proof. 

It suffices to show that the Fourier transform of $P_{T,\mathcal{P}}^{\Omega }$ converges to a certain continuous function. Characters of $\Omega$ have the form

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

where $\omega \left(p\right)$ denotes the pth component of $\omega \in \Omega$, and $k_p$ are integer rational numbers, where only a finite number of them are not zero. Therefore,

$${\mathcal{F}}_{T,\mathcal{P}}\left(\mathbb{k}\right)=\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}}\left({\prod _{p\in \mathcal{P}}}^{\ast }{p}^{-i\tau k_p}\right)\mathrm{d}\tau ,$$

where $\mathbb{k}=(k_p:p\in \mathcal{P})$, and the star ∗ shows that $k_p\ne 0$ for a finite set of generalized primes p, is the Fourier transform of the measure $P_{T,\mathcal{P}}^{\Omega }$. Define two sets of $\mathbb{k}$:

$$K_1=\left\{\mathbb{k}:{\sum _{p\in \mathcal{P}}}^{\ast }{k}_{p}logp=0\right\},K_2=\left\{\mathbb{k}:{\sum _{p\in \mathcal{P}}}^{\ast }{k}_{p}logp\ne 0\right\}.$$

Then, we have

$${\mathcal{F}}_{T,\mathcal{P}}\left(\mathbb{k}\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathbb{k}\in K_1,\hfill \\ \frac{1-exp\left\{-iT{{\sum }^{\ast }}_{p\in \mathcal{P}}{k}_{p}logp\right\}}{iT\left(1-exp\left\{-i{{\sum }^{\ast }}_{p\in \mathcal{P}}{k}_{p}logp\right\}\right)}\hfill & \mathrm{if}\mathbb{k}\in K_2.\hfill \end{array}\right.$$

Thus,

$$\underset{T\to \infty }{lim}{\mathcal{F}}_{T,\mathcal{P}}\left(\mathbb{k}\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathbb{k}\in K_1,\hfill \\ 0\hfill & \mathrm{if}\mathbb{k}\in K_2.\hfill \end{array}\right.$$

The limit function is continuous in the discrete topology; therefore, this implies that $P_{T,\mathcal{P}}^{\Omega }$ weakly converges to the measure $P_{\mathcal{P}}^{\Omega }$ on $(\Omega ,\mathcal{B}(\Omega \left)\right)$ given by the Fourier transform ${\mathcal{F}}_{\mathcal{P}}\left(\mathbb{k}\right)$,

$${\mathcal{F}}_{\mathcal{P}}\left(\mathbb{k}\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathbb{k}\in K_1,\hfill \\ 0\hfill & \mathrm{if}\mathbb{k}\in K_2.\hfill \end{array}\right.$$

□

Remark 1.

If the system $\mathcal{P}$ is linearly independent over the field of rational numbers, then

$${\mathcal{F}}_{\mathcal{P}}\left(\mathbb{k}\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathbb{k}=\left(\mathbf{0}\right),\hfill \\ 0\hfill & \mathrm{if}\mathbb{k}\ne \left(\mathbf{0}\right).\hfill \end{array}\right.$$

In this case, the limit measure $P_{\mathcal{P}}^{\Omega }$ is the Haar measure $P_H$, which is invariant with respect to translations by elements $\omega \in \Omega$, i.e., for every $\omega \in \Omega$ and $A\in \mathcal{B}(\Omega )$,

$$P_H\left(A\right)=P_H\left(\omega A\right)=P_H\left(A\omega \right).$$

Obviously, in this case, the numbers of $\mathcal{P}$ must be different.

Lemma 1 is a starting point to consider limit distributions in space $\mathcal{H}\left(D_{\mathcal{P}}\right)$. The simplest case is of an absolutely convergent Dirichlet series. Let $\eta >1-\widehat{\sigma }$ be fixed. For $m\in {\mathcal{N}}_{\mathcal{P}}$ and $n\in \mathbb{N}$, set

$$a_n\left(m\right)=exp\left\{-{\left(\frac{m}{n}\right)}^{\eta }\right\},$$

and

$${\zeta }_{n,\mathcal{P}}\left(s\right)=\sum _{m\in {\mathcal{N}}_{\mathcal{P}}}\frac{a_n\left(m\right)}{m^s}.$$

It is not difficult to see that the series for ${\zeta }_{n,\mathcal{P}}\left(s\right)$ is absolutely convergent, say, for $\sigma >0$. Thus, ${\zeta }_{n,\mathcal{P}}\left(s\right)$ is an element of $\mathcal{H}\left(D_{\mathcal{P}}\right)$. For $A\in \mathcal{B}\left(\mathcal{H}\left(D_{\mathcal{P}}\right)\right)$, define

$$P_{T,n,\mathcal{P}}\left(A\right)=\frac{1}{T}{m}_L\left\{\tau \in [0,T]:{\zeta }_{\mathcal{P},n}(s+i\tau )\in A\right\}.$$

Lemma 2.

Assume that the system $\mathcal{P}$ satisfies the axiom (1). Then, $P_{T,n,\mathcal{P}}$ weakly converges to a certain measure $P_{n,\mathcal{P}}$ on $(\mathcal{H}\left(D_{\mathcal{P}}\right),\mathcal{B}\left(\mathcal{H}\left(D_{\mathcal{P}}\right)\right))$ as $T\to \infty$.

Proof. 

Extend the function $\omega \left(p\right)$ to the set ${\mathcal{N}}_{\mathcal{P}}$ by using the equality

$$\omega \left(m\right)={\omega }^{a_1}\left(p_1\right)\cdots {\omega }^{a_r}\left(p_r\right)$$

for $m=p_1^{a_1}\cdots p_r^{a_r}$. Consider the mapping $h_{n,\mathcal{P}}:\Omega \to \mathcal{H}\left(D_{\mathcal{P}}\right)$ given by

$$h_{n,\mathcal{P}}\left(\omega \right)=\sum _{m\in {\mathcal{N}}_{\mathcal{P}}}\frac{\omega \left(m\right)a_n\left(m\right)}{m^s},\omega \in \Omega .$$

The latter definition implies that

$$h_{n,\mathcal{P}}\left(p^{-i\tau }:p\in \mathcal{P}\right)=\sum _{m\in {\mathcal{N}}_{\mathcal{P}}}\frac{a_n\left(m\right)}{m^{s+i\tau }}={\zeta }_{n,\mathcal{P}}(s+i\tau ).$$

(6)

Moreover, the absolute convergence of the series

$$\sum _{m\in {\mathcal{N}}_{\mathcal{P}}}\frac{\omega \left(m\right)a_n\left(m\right)}{m^s}$$

for $\sigma >0$ ensures the continuity of the mapping $h_{n,\mathcal{P}}$. In view of (6), we have

$$P_{T,n,\mathcal{P}}\left(A\right)=\frac{1}{T}{m}_L\left\{\tau \in [0,T]:\left(p^{-i\tau }:p\in \mathcal{P}\right)\in h_{n,\mathcal{P}}^{-1}A\right\}=P_{T,\mathcal{P}}^{\Omega }\left(h_{n,\mathcal{P}}^{-1}A\right)$$

for all $A\in \mathcal{B}\left(\mathcal{H}\left(D_{\mathcal{P}}\right)\right)$. This shows that $P_{T,n,\mathcal{P}}=P_{T,\mathcal{P}}^{\Omega }{h}_{n,\mathcal{P}}^{-1}$, where

$$P_{T,\mathcal{P}}^{\Omega }{h}_{n,\mathcal{P}}^{-1}\left(A\right)=P_{T,\mathcal{P}}^{\Omega }\left(h_{n,\mathcal{P}}^{-1}A\right),A\in \mathcal{B}\left(\mathcal{H}\left(D_{\mathcal{P}}\right)\right),$$

and $h_{n,\mathcal{P}}^{-1}A$ denotes the preimage of the set A. These remarks, Lemma 1, and the preservation of weak convergence under continuous mappings (see, for example, [28], Chapter 5) prove that $P_{T,n,\mathcal{P}}$, as $T\to \infty$ weakly converges to the measure $P_{n,\mathcal{P}}=h_{n,\mathcal{P}}^{-1}{P}_{\mathcal{P}}^{\Omega }$, where $P_{\mathcal{P}}^{\Omega }$ is from Lemma 1.  □

3. Some Estimates

To pass from the function ${\zeta }_{n,\mathcal{P}}\left(s\right)$ to ${\zeta }_{\mathcal{P}}\left(s\right)$, we need some estimates between these functions. We start with an integral representation for ${\zeta }_{n,\mathcal{P}}\left(s\right)$. As usual, let $\Gamma \left(s\right)$ stand for the Euler gamma-function, and, for $n\in \mathbb{N}$, define

$$l_n\left(s\right)={\eta }^{-1}\Gamma \left({\eta }^{-1}s\right)n^s,$$

where the number $\eta$ is from the definition of $a_n\left(m\right)$.

Lemma 3.

Suppose that axiom (1) is valid. Then, for $s\in D$, the representation

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

(7)

holds.

Proof. 

Let a and b be positive numbers. Then, the classical Mellin formula

$$\frac{1}{2\pi i}{\displaystyle \underset{a-i\infty }{\overset{a+i\infty }{\int }}}\Gamma \left(z\right)b^{-z}\mathrm{d}z={\mathrm{e}}^{-b}$$

is valid. Therefore, for $m\in {\mathcal{N}}_{\mathcal{P}}$,

$$\frac{1}{2\pi i}{\displaystyle \underset{\eta -i\infty }{\overset{\eta +i\infty }{\int }}}{m}^{-z}{l}_n\left(z\right)\mathrm{d}z=\frac{1}{2\pi i}{\displaystyle \underset{\eta -i\infty }{\overset{\eta +i\infty }{\int }}}\Gamma \left(\frac{z}{\eta }\right){\left(\frac{m}{n}\right)}^{(-z/\eta )\eta }\mathrm{d}\left(\frac{z}{\eta }\right)=a_n\left(m\right).$$

Hence,

$$\begin{array}{cc}\hfill {\zeta }_{n,\mathcal{P}}\left(s\right)& =\sum _{m\in {\mathcal{N}}_{\mathcal{P}}}\frac{a_n\left(m\right)}{m^s}=\frac{1}{2\pi i}\sum _{m\in {\mathcal{N}}_{\mathcal{P}}}{\displaystyle \underset{\eta -i\infty }{\overset{\eta +i\infty }{\int }}}\frac{1}{m^{s+z}}{l}_n\left(z\right)\mathrm{d}z\hfill \\ & =\frac{1}{2\pi i}{\displaystyle \underset{\eta -i\infty }{\overset{\eta +i\infty }{\int }}}\left(\sum _{m\in {\mathcal{N}}_{\mathcal{P}}}\frac{1}{m^{s+z}}\right)l_n\left(z\right)\mathrm{d}z.\hfill \end{array}$$

(8)

Since $\eta >1-\widehat{\sigma }$, we have $\mathrm{Re}(s+z)>1$. Moreover, the properties of the function $\Gamma \left(s\right)$ ensure the change in order integration and summation. Thus, (8) implies the representation of the lemma.  □

There is a sequence of compact embedded sets $\{K_l:l\in \mathbb{N}\}\subset D_{\mathcal{P}}$, $D_{\mathcal{P}}={\displaystyle \underset{l=1}{\overset{\infty }{\cup }}}{K}_l$, such that every compact set $K\subset D_{\mathcal{P}}$ lies in some $K_l$. Then,

$$\rho (g_1,g_2)={\displaystyle \sum _{l=1}^{\infty }}{2}^{-l}\frac{{sup}_{s\in K_l}|g_1\left(s\right)-g_2\left(s\right)|}{1+{sup}_{s\in K_l}|g_1\left(s\right)-g_2\left(s\right)|},g_1,g_2\in \mathcal{H}\left(D_{\mathcal{P}}\right),$$

is a metric in $\mathcal{H}\left(D_{\mathcal{P}}\right)$ inducing its topology of uniform convergence on compacta.

Lemma 4.

Suppose that axiom (1) is valid. Then,

$$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}{\displaystyle \underset{1}{\overset{T}{\int }}}\rho \left({\zeta }_{\mathcal{P}}(s+i\tau ),{\zeta }_{n,\mathcal{P}}(s+i\tau )\right)=0.$$

Proof. 

By the formula for $\rho$, it is sufficient to prove that, for every compact set $K\subset D_{\mathcal{P}}$,

$$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}{\displaystyle \underset{1}{\overset{T}{\int }}}\underset{s\in K}{sup}\left|{\zeta }_{\mathcal{P}}(s+i\tau )-{\zeta }_{n,\mathcal{P}}(s+i\tau )\right|=0.$$

(9)

Thus, fix a compact set $K\subset D_{\mathcal{P}}$. Then, there is $\epsilon >0$ satisfying $\widehat{\sigma }+\epsilon ⩽\sigma ⩽1-\epsilon /2$ for $\sigma +it\in K$. We apply Lemma 3. Let $\eta =1$, and ${\eta }_1=\widehat{\sigma }+\epsilon /2-\sigma$ with above $\sigma$. Then ${\eta }_1<0$. The integrand in (7) possesses a simple pole at $z=0$ (a pole of $\Gamma \left(s\right)$), and a simple pole at $z=1-s$ (a pole of ${\zeta }_{\mathcal{P}}(s+z)$). Actually, it is obvious that $0\in ({\eta }_1,\eta )$ and $1-\sigma \in ({\eta }_1,\eta )$. Moreover, since ${\eta }_1⩾\widehat{\sigma }+\epsilon /2-1+\epsilon /2$, $\widehat{\sigma }-1+\epsilon >-1$, the pole $z=-1$ of $\Gamma \left(s\right)$ does not lie in the strip ${\eta }_1<\mathrm{Re}z<\eta$.

Now, the residue theorem and Lemma 3 yields, for $s\in K$,

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

Hence, for $s\in K$,

$$\begin{array}{cc}\hfill & {\zeta }_{n,\mathcal{P}}(s+i\tau )-{\zeta }_{\mathcal{P}}(s+i\tau )\hfill \\ & =\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}{\zeta }_{\mathcal{P}}\left(\widehat{\sigma }+\frac{\epsilon }{2}+i\tau +it+iu\right)l_n\left(\widehat{\sigma }+\frac{\epsilon }{2}-\sigma +iu\right)\mathrm{d}u+a{l}_n(1-s-i\tau )\hfill \\ & =\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}{\zeta }_{\mathcal{P}}\left(\widehat{\sigma }+\frac{\epsilon }{2}+i\tau +iu\right)l_n\left(\widehat{\sigma }+\frac{\epsilon }{2}-s+iu\right)\mathrm{d}u+a{l}_n(1-s-i\tau )\hfill \\ & \ll {\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}\left|{\zeta }_{\mathcal{P}}\left(\widehat{\sigma }+\frac{\epsilon }{2}+i\tau +iu\right)\right|\underset{s\in K}{sup}\left|l_n\left(\widehat{\sigma }+\frac{\epsilon }{2}-s+iu\right)\right|\mathrm{d}u+\underset{s\in K}{sup}\left|l_n(1-s-i\tau )\right|.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill \frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}}& \underset{s\in K}{sup}\left|{\zeta }_{\mathcal{P}}(s+i\tau )-{\zeta }_{n,\mathcal{P}}(s+i\tau )\right|\mathrm{d}\tau \hfill \\ \hfill \ll & {\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}\left(\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}}\left|{\zeta }_{\mathcal{P}}\left(\widehat{\sigma }+\frac{\epsilon }{2}+i\tau +iu\right)\right|\mathrm{d}\tau \right)\underset{s\in K}{sup}\left|l_n(1-s+iu)\right|\mathrm{d}u\hfill \\ & +\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}}\underset{s\in K}{sup}\left|l_n(1-s-i\tau )\right|\mathrm{d}\tau \hfill \\ \hfill \stackrel{\mathrm{def}}{=}& J_1+J_2.\hfill \end{array}$$

(10)

By the definition of $\widehat{\sigma }$,

$${\displaystyle \underset{0}{\overset{T}{\int }}}{\left|{\zeta }_{\mathcal{P}}\left(\widehat{\sigma }+\frac{\epsilon }{2}+i\tau \right)\right|}^2\mathrm{d}\tau {\ll }_{\epsilon }T.$$

Therefore, in view of the Cauchy–Schwarz inequality,

$$\begin{array}{cc}\hfill {\displaystyle \underset{0}{\overset{T}{\int }}}\left|{\zeta }_{\mathcal{P}}\left(\widehat{\sigma }+\frac{\epsilon }{2}+i\tau +iu\right)\right|\mathrm{d}\tau & ⩽\sqrt{T}{\left({\displaystyle \underset{0}{\overset{T}{\int }}}{\left|{\zeta }_{\mathcal{P}}\left(\widehat{\sigma }+\frac{\epsilon }{2}+i\tau +iu\right)\right|}^2\mathrm{d}\tau \right)}^{1/2}\hfill \\ & ⩽\sqrt{T}{\left({\displaystyle \underset{-\left|u\right|}{\overset{T+\left|u\right|}{\int }}}{\left|{\zeta }_{\mathcal{P}}\left(\widehat{\sigma }+\frac{\epsilon }{2}+i\tau \right)\right|}^2\mathrm{d}\tau \right)}^{1/2}\hfill \\ & {\ll }_{\epsilon }\sqrt{T}{(T+|u\left|\right)}^{1/2}{\ll }_{\epsilon }\sqrt{T}\left(\sqrt{T}+\sqrt{u}\right)\hfill \\ & {\ll }_{\epsilon }T\left(1+\sqrt{u}\right).\hfill \end{array}$$

(11)

The most important ingredient of the function $l_n\left(s\right)$ is $\Gamma \left(s\right)$ and is estimated as

$$\Gamma (\sigma +it)\ll exp\{-c|t\left|\right\},c>0.$$

Therefore, for $s\in K$,

$$l_n\left(\widehat{\sigma }+\frac{\epsilon }{2}+1-s+iu\right)\ll n^{\widehat{\sigma }+\epsilon /2-\sigma }exp\{-c|u-t\left|\right\}{\ll }_K{n}^{-\epsilon /2}exp\{-c_1\left|u\right|\},c_1>0.$$

This, together with (11), yields

$$J_1{\ll }_{K,\epsilon }{n}^{-\epsilon /2}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}\left(1+\sqrt{u}\right)exp\{-c_1\left|u\right|\}\mathrm{d}u{\ll }_{\epsilon ,K}{n}^{-\epsilon /2}.$$

(12)

Similarly, as above, we obtain that, for $s\in K$,

$$l_n(1-s-i\tau )\ll n^{1-\sigma }exp\{-c|t+\tau \left|\right\}{\ll }_K{n}^{1-\widehat{\sigma }-\epsilon }exp\{-c_2\left|\tau \right|\},c_2>0.$$

Therefore,

$$J_2{\ll }_K{n}^{1-\widehat{\sigma }-\epsilon }\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}}exp\{-c_2\left|\tau \right|\}\mathrm{d}\tau {\ll }_K{n}^{1-\widehat{\sigma }-\epsilon }{T}^{-1}.$$

The latter bound, (12) and (10), prove (9). The lemma is proved.  □

4. Proof of Theorem 2

We derive Theorem 2 from Lemmas 2 and 4 and the following statement (see, for example, [28], Theorem 4.2) is applied to the case $\mathcal{H}\left(D_{\mathcal{P}}\right)$.

Lemma 5.

Assume that ${\xi }_{nk}$ and ${\widehat{\xi }}_n$, $n,k\in \mathbb{N}$, are $\mathcal{H}\left(D_{\mathcal{P}}\right)$-valued random elements given on a space $(\mathbb{X},\mathcal{B}(\mathbb{X}),\nu )$. Let

$${\xi }_{nk}{\displaystyle \underset{n\to \infty }{\overset{\mathcal{D}}{\to }}}{\xi }_k,{\xi }_k{\displaystyle \underset{k\to \infty }{\overset{\mathcal{D}}{\to }}}\xi ,$$

and for $\epsilon >0$,

$$\underset{k\to \infty }{lim}\underset{n\to \infty }{lim\; sup}\nu \left\{\rho \left({\widehat{\xi }}_n,{\xi }_{nk}\right)⩾\epsilon \right\}=0,$$

where ${\displaystyle \stackrel{\mathcal{D}}{\to }}$ stands for the convergence in distribution. Then ${\widehat{\xi }}_n{\displaystyle \underset{n\to \infty }{\overset{\mathcal{D}}{\to }}}\xi$.

We remind the reader that $P_{n,\mathcal{P}}$ is from Lemma 2. Using Lemma 5 requires some convergence properties for $P_{n,\mathcal{P}}$. Recall that the sequence $\{P_{n,\mathcal{P}}:n\in \mathbb{N}\}$ is tight if, for every $\epsilon >0$, there is a compact set $K\subset \mathcal{H}\left(D_{\mathcal{P}}\right)$ such that

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

with all $n\in \mathbb{N}$.

Lemma 6.

Suppose that the system $\mathcal{P}$ satisfies the axiom (1). Then, the sequence $\{P_{n,\mathcal{P}}:n\in \mathbb{N}\}$ is tight.

Proof. 

Let $K_l$ be a fixed compact set in the definition of $\rho$. Then, the Cauchy integral theorem, for $s\in K_l$, implies

$${\zeta }_{\mathcal{P}}(s+i\tau )=\frac{1}{2\pi i}\underset{\mathcal{L}}{\int }\frac{{\zeta }_{\mathcal{P}}(z+i\tau )}{z-s}\mathrm{d}z,$$

where $\mathcal{L}$ is a closed simple curve lying in D and enclosing the set $K_l$. Hence,

$$\underset{s\in K_l}{sup}{\left|{\zeta }_{\mathcal{P}}(s+i\tau )\right|}^2\ll \underset{\mathcal{L}}{\int }\frac{\left|\mathrm{d}z\right|}{{|z-s|}^2}\underset{\mathcal{L}}{\int }{\left|{\zeta }_{\mathcal{P}}(z+i\tau )\right|}^2\left|\mathrm{d}z\right|{\ll }_{K_l}\underset{\mathcal{L}}{\int }{\left|{\zeta }_{\mathcal{P}}(\mathrm{Re}z+i\mathrm{Im}z+i\tau )\right|}^2\left|\mathrm{d}z\right|.$$

Therefore,

$$\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}}\underset{s\in K_l}{sup}{\left|{\zeta }_{\mathcal{P}}(s+i\tau )\right|}^2\mathrm{d}\tau {\ll }_{K_l}\underset{\mathcal{L}}{\int }\left(\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}}{\left|{\zeta }_{\mathcal{P}}(\mathrm{Re}z+i\mathrm{Im}z+i\tau )\right|}^2\mathrm{d}\tau \right)\left|\mathrm{d}z\right|{\ll }_{K_l}1⩽B_l<\infty .$$

From this, we have

$$\underset{T\to \infty }{lim\; sup}\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}}\underset{s\in K_l}{sup}\left|{\zeta }_{\mathcal{P}}(s+i\tau )\right|\mathrm{d}\tau ⩽\sqrt{B_l}.$$

Then, in view of (9),

$$\begin{array}{cc}\hfill \underset{n\in \mathbb{N}}{sup}\underset{T\to \infty }{lim\; sup}\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}}\underset{s\in K_l}{sup}\left|{\zeta }_{n,\mathcal{P}}(s+i\tau )\right|\mathrm{d}\tau ⩽& \underset{n\in \mathbb{N}}{sup}\underset{T\to \infty }{lim\; sup}\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}}\underset{s\in K_l}{sup}\left|{\zeta }_{\mathcal{P}}(s+i\tau )-{\zeta }_{n,\mathcal{P}}(s+i\tau )\right|\mathrm{d}\tau \hfill \\ & +\underset{T\to \infty }{lim\; sup}\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}}\underset{s\in K_l}{sup}\left|{\zeta }_{\mathcal{P}}(s+i\tau )\right|\mathrm{d}\tau \hfill \\ \hfill ⩽& C_l<\infty .\hfill \end{array}$$

(13)

Let ${\beta }_T$ be the random variable on the space $(\widehat{\Omega },\mathcal{A},\nu )$ and uniformly distributed in $[0,T]$. Define $\mathcal{H}\left(D_{\mathcal{P}}\right)$-valued random elements

$${\xi }_{T,n}={\xi }_{T,n}\left(s\right)={\zeta }_{n,\mathcal{P}}(s+i{\beta }_T)$$

and ${\xi }_n={\xi }_n\left(s\right)$ having the distribution $P_{n,\mathcal{P}}$. We fix $\epsilon >0$, and set $V=V_l=2^{-l}{\epsilon }^{-1}{C}_l$. Then, in virtue of (13) and Lemma 2,

$$\begin{array}{cc}\hfill \nu \left\{\underset{s\in K_l}{sup}\left|{\xi }_n\left(s\right)\right|⩾V_l\right\}& ⩽\underset{T\to \infty }{lim\; sup}\nu \left\{\underset{s\in K_l}{sup}\left|{\xi }_{T,n}\left(s\right)\right|⩾V_l\right\}\hfill \\ & ⩽\underset{n\in \mathbb{N}}{sup}\underset{T\to \infty }{lim\; sup}\frac{1}{V_l}{\displaystyle \underset{0}{\overset{T}{\int }}}\underset{s\in K_l}{sup}\left|{\zeta }_{n,\mathcal{P}}(s+i\tau )\right|\mathrm{d}\tau =\frac{\epsilon }{2^l}\hfill \end{array}$$

(14)

for all $n\in \mathbb{N}$. Let $K=\left\{h\in \mathcal{H}\left(D_{\mathcal{P}}\right):{sup}_{s\in K_l}\left|h\left(s\right)\right|⩽V_l,l\in \mathbb{N}\right\}$. Then, K is a compact set in $\mathcal{H}\left(D_{\mathcal{P}}\right)$, and, by (14),

$$\begin{array}{cc}\hfill P_{n,\mathcal{P}}\left(K\right)& =1-P_{n,\mathcal{P}}(\mathcal{H}\left(D_{\mathcal{P}}\right)\setminus K)=1-P_{n,\mathcal{P}}\left(g\left(s\right)\in \mathcal{H}\left(D_{\mathcal{P}}\right):\exists l:\underset{s\in K_l}{sup}\left|g\left(s\right)\right|⩾V_l\right)\hfill \\ & =1-P_{n,\mathcal{P}}\left({\displaystyle \bigcup _{l=1}^{\infty }}\left\{g\left(s\right)\in \mathcal{H}\left(D_{\mathcal{P}}\right):\underset{s\in K_l}{sup}\left|g\left(s\right)\right|⩾V_l\right\}\right)\hfill \\ & ⩾1-{\displaystyle \sum _{l=1}^{\infty }}{P}_{n,\mathcal{P}}\left(g\left(s\right)\in \mathcal{H}\left(D_{\mathcal{P}}\right):\underset{s\in K_l}{sup}|g\left(s\right)⩾V_l\right)\hfill \\ & =1-{\displaystyle \sum _{l=1}^{\infty }}\nu \left\{\underset{s\in K_l}{sup}\left|{\xi }_n\left(s\right)\right|⩾V_l\right\}⩾1-\epsilon {\displaystyle \sum _{l=1}^{\infty }}{2}^{-l}=1-\epsilon \hfill \end{array}$$

for all $n\in \mathbb{N}$. This proves the lemma.  □

Proof of Theorem 2.

We will apply Lemma 5. Since by Lemma 6, the sequence $\{P_{n,\mathcal{P}}:n\in \mathbb{N}\}$ is tight, it is relatively compact in virtue of the classical Prokhorov theorem; see, for example, [28], Theorem 6.1. This means that every subsequence of $\left\{P_{n,\mathcal{P}}\right\}$ possesses a subsequent weak convergent to a probability measure on $(\mathcal{H}\left(D_{\mathcal{P}}\right),\mathcal{B}\left(\mathcal{H}\left(D_{\mathcal{P}}\right)\right))$. Thus, there is $\left\{P_{n_r,\mathcal{P}}\right\}\subset \left\{P_{n,\mathcal{P}}\right\}$ and a probability measure $P_{\mathcal{P}}$ on $(\mathcal{H}\left(D_{\mathcal{P}}\right),\mathcal{B}\left(\mathcal{H}\left(D_{\mathcal{P}}\right)\right))$ such that $P_{n_r,\mathcal{P}}$ converges weakly to $P_{\mathcal{P}}$ as $r\to \infty$. Using the notation of the proof of Lemma 6, we have

$${\xi }_{n_r}{\displaystyle \underset{r\to \infty }{\overset{\mathcal{D}}{\to }}}{P}_{\mathcal{P}}.$$

(15)

Moreover, in view of Lemma 2,

$${\xi }_{T,n}{\displaystyle \underset{T\to \infty }{\overset{\mathcal{D}}{\to }}}{\xi }_n.$$

(16)

Define one more $\mathcal{H}\left(D_{\mathcal{P}}\right)$-valued random element

$${\widehat{\xi }}_T={\widehat{\xi }}_T\left(s\right)={\zeta }_{\mathcal{P}}(s+i{\beta }_T).$$

Then Lemma 4 implies that, for every $\epsilon >0$,

$$\begin{array}{cc}\hfill \underset{r\to \infty }{lim}& \underset{T\to \infty }{lim\; sup}\nu \left\{\rho \left({\widehat{\xi }}_T,{\xi }_{T,n}\right)⩾\epsilon \right\}\hfill \\ & =\underset{r\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}{m}_L\left\{\tau \in [0,T]:\rho \left({\zeta }_{\mathcal{P}}(s+i\tau ),{\zeta }_{n_r,\mathcal{P}}(s+i\tau )\right)⩾\epsilon \right\}\hfill \\ & ⩽\underset{r\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{\epsilon T}{\displaystyle \underset{0}{\overset{T}{\int }}}\rho \left({\zeta }_{\mathcal{P}}(s+i\tau ),{\zeta }_{n_r,\mathcal{P}}(s+i\tau )\right)\mathrm{d}\tau =0.\hfill \end{array}$$

This equality, together with (15) and (16), shows that for ${\xi }_{n_r}$, ${\xi }_{T,n}$ and ${\widehat{\xi }}_T$, the conditions of Lemma 5 are fulfilled. Therefore, the relation

$${\widehat{\xi }}_T{\displaystyle \underset{T\to \infty }{\overset{\mathcal{D}}{\to }}}{P}_{\mathcal{P}}$$

holds, and this implies the weak convergence of $P_{T,\mathcal{P}}$ to $P_{\mathcal{P}}$ as $T\to \infty$. The proof is completed.  □

5. Proof of Theorem 1

Theorem 1 is a consequence of Theorem 2 and the equivalents of weak convergence.

We remind the reader that the support of the measure $P_{\mathcal{P}}$ is a closed minimal set $S_{\mathcal{P}}\subset \mathcal{H}\left(D_{\mathcal{P}}\right)$ satisfying $P_{\mathcal{P}}\left(S_{\mathcal{P}}\right)=1$. The set $S_{\mathcal{P}}$ contains all $g\in \mathcal{H}\left(D_{\mathcal{P}}\right)$ such that for any open neighborhood $\mathcal{G}$ of g, the inequality $P_{\mathcal{P}}\left(\mathcal{G}\right)>0$ holds.

Proof of Theorem 1.

Let $F_{\mathcal{P}}$ be the support of the limit measure $P_{\mathcal{P}}$ in Theorem 2. Then, $F_{\mathcal{P}}$ is a closed set, and $F_{\mathcal{P}}\ne ⌀$ because $P_{\mathcal{P}}\left(F_{\mathcal{P}}\right)=1$. We will prove that the set $F_{\mathcal{P}}$ has approximation properties of the theorem.

Suppose that $f\left(s\right)\in F_{\mathcal{P}}$, and

$${\mathcal{G}}_{\epsilon }=\left\{h\in \mathcal{H}\left(D_{\mathcal{P}}\right):\underset{s\in K}{sup}|h\left(s\right)-f\left(s\right)|<\epsilon \right\},$$

i.e., ${\mathcal{G}}_{\epsilon }$ is an open neighborhood of an element $f\left(s\right)$ of the support $F_{\mathcal{P}}$. Hence, by the support property,

$$P_{\mathcal{P}}\left({\mathcal{G}}_{\epsilon }\right)>0.$$

(17)

Moreover, using Theorem 2 and Theorem 2.1 of [28] with open sets implies the inequality

$$\underset{T\to \infty }{lim\; inf}{P}_{T,\mathcal{P}}\left({\mathcal{G}}_{\epsilon }\right)⩾P_{\mathcal{P}}\left({\mathcal{G}}_{\epsilon }\right).$$

Thus, the notations for $P_{T,\mathcal{P}}$ and ${\mathcal{G}}_{\epsilon }$ lead to

$$\underset{T\to \infty }{lim\; inf}\frac{1}{T}{m}_L\left\{\tau \in [0,T]:\underset{s\in K}{sup}\left|{\zeta }_{\mathcal{P}}(s+i\tau )-f\left(s\right)\right|<\epsilon \right\}>0.$$

To prove the second statement of the theorem, we deal with continuity sets. We remind the reader that a set $A\in \mathcal{B}\left(\mathbb{X}\right)$ is a continuity set of a measure P on $(\mathbb{X},\mathcal{B}(\mathbb{X}\left)\right)$ if $P(\partial A)=0$, where $\partial A$ is the boundary of A.

The set $\partial {\mathcal{G}}_{\epsilon }$ of the set ${\mathcal{G}}_{\epsilon }$ belongs to the set

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

Hence, the sets $\partial {\mathcal{G}}_{{\epsilon }_1}$ and $\partial {\mathcal{G}}_{{\epsilon }_2}$ for different ${\epsilon }_1$ and ${\epsilon }_2$ have no common elements. From this remark, it follows that $P_{\mathcal{P}}(\partial {\mathcal{G}}_{\epsilon })>0$ for at most countably many values of $\epsilon$, or, in the above terminology, the set ${\mathcal{G}}_{\epsilon }$ is a continuity set of the measure $P_{\mathcal{P}}$ for all but at most countably many $\epsilon >0$. Thus, Theorem 2 and Theorem 2.1 of [28] with continuity sets show that the limit

$$\underset{T\to \infty }{lim}{P}_{T,\mathcal{P}}\left({\mathcal{G}}_{\epsilon }\right)=P_{\mathcal{P}}\left({\mathcal{G}}_{\epsilon }\right)$$

exists, and in view of (17), is positive for all but at most countably many $\epsilon >0$. This and the notations for $P_{T,\mathcal{P}}$ and ${\mathcal{G}}_{\epsilon }$ give the second assertion of the theorem. The theorem is proved.  □

6. Conclusions

Every system $\mathcal{P}$ of real numbers $1<p_1⩽p_2⩽\cdots ⩽p_r⩽\cdots$, ${lim}_{n\to \infty }{p}_n=\infty$ is called generalized prime numbers. We consider the zeta-function ${\zeta }_{\mathcal{P}}\left(s\right)$, $s=\sigma +it$ associated with the system $\mathcal{P}$. We assume that the system of generalized integers ${\mathcal{N}}_{\mathcal{P}}$ obtained from $\mathcal{P}$ satisfies the axiom

$$\underset{m\in \mathcal{N}}{\sum _{m⩽x}}1=ax+O\left(x^{\delta }\right),a>0,0⩽\delta <1.$$

Then, for $\sigma >1$, the function ${\zeta }_{\mathcal{P}}\left(s\right)$ is defined by

$${\zeta }_{\mathcal{P}}\left(s\right)=\sum _{m\in \mathcal{N}}\frac{1}{m^s}=\prod _{p\in \mathcal{P}}{\left(1-\frac{1}{p^s}\right)}^{-1},$$

and has analytic continuation to the region $\delta <\sigma <1$. Additionally, we suppose that ${\zeta }_{\mathcal{P}}\left(s\right)$ has the bounded mean square

$${\int }_0^T{\left|{\zeta }_{\mathcal{P}}(\sigma +it)\right|}^2\mathrm{d}t{\ll }_{\sigma }T,T\to \infty ,$$

for some $\sigma >\widehat{\sigma }$ with some $\delta <\widehat{\sigma }<1$.

We consider probabilistic and approximation properties of the function ${\zeta }_{\mathcal{P}}\left(s\right)$. We prove a limit theorem for ${\zeta }_{\mathcal{P}}\left(s\right)$ in the space of analytic functions $\mathcal{H}\left(D_{\mathcal{P}}\right)$, $D_{\mathcal{P}}=\{s\in \mathbb{C}:\widehat{\sigma }<\sigma <1\}$, i.e., that

$$\frac{1}{T}{m}_L\left\{\tau \in [0,T]:{\zeta }_{\mathcal{P}}(s+i\tau )\in A\right\},A\in \mathcal{B}\left(H\left(D_{\mathcal{P}}\right)\right),$$

converges weakly to a certain probability measure $P_{\mathcal{P}}$ as $T\to \infty$. From this, we deduce that the shifts ${\zeta }_{\mathcal{P}}(s+i\tau )$ approximate a certain closed subset of $\mathcal{H}\left(D_{\mathcal{P}}\right)$.

For identification of the limit measure $P_{\mathcal{P}}$ and universality of the function ${\zeta }_{\mathcal{P}}\left(s\right)$, some stronger restrictions for the system $\mathcal{P}$ are needed. We are planning to apply this in the future.