On Universality of Some Beurling Zeta-Functions

Abstract

Let $\mathcal{P}$ be the set of generalized prime numbers, and ${\zeta }_{\mathcal{P}}\left(s\right)$, $s=\sigma +it$, denote the Beurling zeta-function associated with $\mathcal{P}$. In the paper, we consider the approximation of analytic functions by using shifts ${\zeta }_{\mathcal{P}}(s+i\tau )$, $\tau \in \mathbb{R}$. We assume the classical axioms for the number of generalized integers and the mean of the generalized von Mangoldt function, the linear independence of the set $\{logp:p\in \mathcal{P}\}$, and the existence of a bounded mean square for ${\zeta }_{\mathcal{P}}\left(s\right)$. Under the above hypotheses, we obtain the universality of the function ${\zeta }_{\mathcal{P}}\left(s\right)$. This means that the set of shifts ${\zeta }_{\mathcal{P}}(s+i\tau )$ approximating a given analytic function defined on a certain strip $\widehat{\sigma }<\sigma <1$ has a positive lower density. This result opens a new chapter in the theory of Beurling zeta functions. Moreover, it supports the Linnik–Ibragimov conjecture on the universality of Dirichlet series. For the proof, a probabilistic approach is applied.

1. Introduction

A positive integer $q>1$ is called prime if it has only two divisors, q and 1. Thus, $2,3,5,7,11,\cdots$ are prime numbers. Integer numbers $k>1$ that have divisors different from k and 1 are called composite. It is well known that the set of all primes is infinite, and this was first proved by Euclid. By the fundamental theorem of arithmetic, every integer $k>1$ has a unique representation as a product of prime numbers. Thus,

$$k=q_1^{{\alpha }_1}\cdots q_r^{{\alpha }_r},{\alpha }_j\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\},$$

and $q_j$ is the jth prime number, $j=1,\cdots ,r$, with some $r\in \mathbb{N}$.

Investigations of the number of prime numbers

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

were more complicated. We recall that $a=O\left(b\right)$, $a\in \mathbb{C}$, $b>0$, means that there exists a constant $c>0$ such that $\left|a\right|⩽cb$. Comparatively recently, in 1896 Hadamard [1] and de la Vallée-Poussin [2] proved independently the asymptotic formula

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

For this, they applied the Riemann idea [3] of using the function

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

now called the Riemann zeta-function. The distribution low of prime numbers was found.

Prime numbers have generalizations. The system $\mathcal{P}$ of real numbers $1<p_1⩽p_2⩽\cdots ⩽p_n⩽\cdots$ such that ${lim}_{n\to \infty }{p}_n=\infty$ are called generalized prime numbers. Generalized prime numbers were introduced by Beurling in [4], and are studied by many authors. The system $\mathcal{P}$ generates the associated system ${\mathcal{N}}_{\mathcal{P}}$ of generalized integers consisting of finite products of the form

$$p_1^{{\alpha }_1}\cdots p_r^{{\alpha }_r},{\alpha }_j\in {\mathbb{N}}_0,j=1,\cdots r,$$

with some $r\in \mathbb{N}$.

The main problem in the theory of generalized primes is the asymptotic behavior of the function

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

The function ${\pi }_{\mathcal{P}}\left(x\right)$ is closely connected to the number of generalized integers

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

In these definitions, the sums are taking counting multiplicities of p and m. Distribution results for generalized numbers were obtained by Beurling [4], Borel [5], Diamond [6,7,8], Malvin [9], Nyman [10], Ryavec [11], Hilberdink and Lapidus [12], Stankus [13], Zhang [14], and others. The important place in generalized number theory is devoted to making relations between ${\mathcal{N}}_{\mathcal{P}}\left(x\right)$ and ${\pi }_{\mathcal{P}}\left(x\right)$. We mention some of them. From a general Landau’s theorem for prime ideals [15], we have the estimate

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

(1)

that implies

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

Nyman proved [10] that the estimates

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

(2)

and

$${\pi }_{\mathcal{P}}\left(x\right)=\underset{2}{\overset{x}{\int }}\frac{\mathrm{d}u}{logu}+O\left(\frac{x}{{(logx)}^{{\alpha }_1}}\right),{\alpha }_1>0,$$

with arbitrary $\alpha >0$ and ${\alpha }_1>0$ are equivalent. Beurling observed [4] that the relation

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

is implied by (2) with $\alpha >3/2$.

It is important to stress that Beurling began to use zeta-functions for investigations of the function ${\pi }_{\mathcal{P}}\left(x\right)$. These zeta-functions ${\zeta }_{\mathcal{P}}\left(s\right)$, now called Beurling zeta-functions, are defined in some half-plane $\sigma >{\sigma }_0$, 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

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

where ${\sigma }_0$ depends on the system $\mathcal{P}$.

Suppose that (1) is true. Then, the partial summation shows that the series for ${\zeta }_{\mathcal{P}}\left(s\right)$ is absolutely convergent for $\sigma >1$,

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

(3)

the function ${\zeta }_{\mathcal{P}}\left(s\right)$ is analytic for $\sigma >1$, and the equality

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

is valid.

Analytic continuation for the function ${\zeta }_{\mathcal{P}}\left(s\right)$ is not an easy problem. If (1) is true, then (3) implies

$${\zeta }_{\mathcal{P}}\left(s\right)=\frac{as}{s-1}+s\underset{1}{\overset{\infty }{\int }}\frac{R\left(x\right)}{x^{s+1}}\mathrm{d}x,R\left(x\right)=O\left(x^{\beta }\right),0⩽\beta <1.$$

This gives analytic continuation for ${\zeta }_{\mathcal{P}}\left(s\right)$ to the half-plane $\sigma >\beta$, except for the point $s=1$ which is a simple pole with residue a.

Beurling zeta-functions are attractive analytic objects; investigations of their properties lead to interesting results, and require new methods. Various authors put much effort into showing that the Beurling zeta-functions have similar properties to classical ones. We mention a recent paper [16] containing deep zero-distribution results for ${\zeta }_{\mathcal{P}}\left(s\right)$.

In this paper, we investigate the analytic properties of the function ${\zeta }_{\mathcal{P}}\left(s\right)$. The approximation of analytic functions is one of the most important chapters of function theory. It is well known that the Riemann zeta-function $\zeta \left(s\right)$ is universal in the sense of approximation of analytic functions. More precisely, this means that every non-vanishing analytic function defined on the strip $\{s\in \mathbb{C}:1/2<\sigma <1\}$ can be approximated with desired accuracy by using shifts $\zeta (s+i\tau )$, $\tau \in \mathbb{R}$. Universality of $\zeta \left(s\right)$ and other zeta-functions has deep theoretical (zero-distribution, functional independence, set denseness, moment problem, ...) and practical (approximation problem, quantum mechanics) applications. On the other hand, the universality theory of zeta-functions has some interior problems (effectivization, description of a class of universal functions, Linnik–Ibragimov conjecture, see Section 1.6 of [17], ...); therefore, investigations of universality are continued, see [17,18,19,20,21,22,23].

Our purpose is to prove the universality of the function ${\zeta }_{\mathcal{P}}\left(s\right)$ with a certain system $\mathcal{P}$. We began studying the approximation of analytic functions by shifts ${\zeta }_{\mathcal{P}}(s+i\tau )$ in [24]. Suppose that the estimate (1) is valid. Let

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

$$\widehat{\sigma }=inf\left\{\sigma :M_{\mathcal{P}}(\sigma ,T){\ll }_{\sigma }T,\sigma >max\left(\frac{1}{2},\beta \right)\right\}.$$

Suppose that $\widehat{\sigma }<1$ and define

$$D=D_{\mathcal{P}}=\{s\in \mathbb{C}:\widehat{\sigma }<\sigma <1\}.$$

Here, and in the sequel, the notation $a{\ll }_{c}b$, $a\in \mathbb{C}$, $b>0$, shows that there exists a constant $c=c\left(\epsilon \right)>0$ such that $\left|a\right|⩽cb$. Denote by $H\left(D\right)$ the space of analytic on D functions equipped with the topology of uniform convergence on compacta, and by $\mathrm{meas}A$ the Lebesgue measure of a measurable set $A\subset \mathbb{R}$. The main result of [24] is the following theorem.

Theorem 1.

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

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

Moreover, the limit

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

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

Theorem 1 demonstrates good approximation properties of the function ${\zeta }_{\mathcal{P}}\left(s\right)$; however, the set $F_{\mathcal{P}}$ of approximated functions is not explicitly given. The aim of this paper, using certain additional information on system $\mathcal{P}$, is to identify the set $F_{\mathcal{P}}$.

A new approach for analytic continuation of the function ${\zeta }_{\mathcal{P}}\left(s\right)$ involving the generalized von Mangoldt 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

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

was proposed in [12]. Let, for $\alpha \in [0,1)$ and every $\epsilon >0$,

$${\psi }_{\mathcal{P}}\left(x\right)=x+O\left(x^{\alpha +\epsilon }\right).$$

(4)

Then, in [12], it was obtained that the function ${\zeta }_{\mathcal{P}}\left(s\right)$ is analytic in the half-plane $\sigma >\alpha$, except for a simple pole at the point $s=1$. It turns out that estimates of type (4) are useful for the characterization of the system $\mathcal{P}$. It is known [12] that (1) does not imply the estimate

$${\psi }_{\mathcal{P}}\left(x\right)=x+O\left(x^{{\beta }_1}\right)$$

(5)

with ${\beta }_1<1$. Therefore, together with (1), we suppose that estimate (5) is valid.

Let $\mathcal{K}$ be the class of compact subsets of strip D with the connected complement, and $H_0\left(K\right)$ with $K\in \mathcal{K}$ the class of continuous functions on K that are analytic in the interior of K. Moreover, let

$$L\left(\mathcal{P}\right)=\{logp:p\in \mathcal{P}\}.$$

Note, that the following theorem supports the Linnik–Ibragimov conjecture.

Theorem 2.

Suppose that the system $\mathcal{P}$ satisfies the axioms (1) and (5), and $L\left(\mathcal{P}\right)$ is linearly independent over the field of rational numbers $\mathbb{Q}$. Let $K\in \mathcal{K}$ and $f\left(s\right)\in H_0\left(K\right)$. Then, for every $\epsilon >0$,

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

Moreover, the limit

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

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

Notice that the requirement on the set $L\left(\mathcal{P}\right)$ is sufficiently strong, it shows that the numbers of the system $\mathcal{P}$ must be different. The simplest example is the system

$$\mathcal{P}=\{q+\alpha :q\mathrm{is}\mathrm{prime}\},$$

where $\alpha$ is a transcendental number.

An example of $\mathcal{P}$ with a bounded mean square is given in [25].

For the proof of Theorem 2, we will build the probabilistic theory of the function ${\zeta }_{\mathcal{P}}\left(s\right)$ in the space of analytic functions $H\left(D\right)$.

The paper is organized as follows. In Section 2, we introduce a certain probability space, and define the $H\left(D\right)$ valued random element. Section 3 is devoted to the ergodicity of one group of transformations. In Section 4, we approximate the mean of the function ${\zeta }_{\mathcal{P}}\left(s\right)$ by an absolutely convergent Dirichlet series. Section 5 is the most important. In this section, we prove a probabilistic limit theorem for the function ${\zeta }_{\mathcal{P}}\left(s\right)$ on a weakly convergent probability measure in the space $H\left(D\right)$, and identify the limit measure. Section 6 gives the explicit form for the support of the limit measure of Section 5. In Section 7, the universality of the function ${\zeta }_{\mathcal{P}}\left(s\right)$ is proved.

2. Random Element

Define the Cartesian product

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

The set ${\mathsf{\Omega }}_{\mathcal{P}}$ consists of all functions $\omega :\mathcal{P}\to \{s\in \mathbb{C}:|s|=1\}$. In ${\mathsf{\Omega }}_{\mathcal{P}}$, the operation of pointwise multiplication and product topology can be defined, and this makes ${\mathsf{\Omega }}_{\mathcal{P}}$ a topological group. Since the unit circle is a compact set, the group ${\mathsf{\Omega }}_{\mathcal{P}}$ is compact. Denote by $\mathcal{B}\left(\mathbb{X}\right)$, the Borel $\sigma$-field of the space $\mathbb{X}$. Then, the compactness of ${\mathsf{\Omega }}_{\mathcal{P}}$ implies the existence of the probability Haar measure $m_{\mathcal{P}}$ on $({\mathsf{\Omega }}_{\mathcal{P}},\mathcal{B}\left({\mathsf{\Omega }}_{\mathcal{P}}\right))$, and we have the probability space $({\mathsf{\Omega }}_{\mathcal{P}},\mathcal{B}\left({\mathsf{\Omega }}_{\mathcal{P}}\right),m_{\mathcal{P}})$.

Denote the elements of ${\mathsf{\Omega }}_{\mathcal{P}}$ by $\omega =\left(\omega \right(p):p\in \mathcal{P})$. Since the Haar measure $m_{\mathcal{P}}$ is the product of Haar measures on unit circles, $\left\{\omega \right(p):p\in \mathcal{P}\}$ is a sequence of independent complex-valued random variables uniformly distributed on the unit circle.

Extend the functions $\omega \left(p\right)$, $p\in \mathcal{P}$, to the generalized integers ${\mathcal{N}}_{\mathcal{P}}$. Let

$$m=p_1^{{\alpha }_1}\cdots p_r^{{\alpha }_r}\in {\mathcal{N}}_{\mathcal{P}}.$$

Then we put

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

(6)

Now, for $s\in D$ and $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$, define

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

Lemma 1.

Under the hypotheses of Theorem 2, ${\zeta }_{\mathcal{P}}(s,\omega )$ is an $H\left(D\right)$-valued random element defined on the probability space $({\mathsf{\Omega }}_{\mathcal{P}},\mathcal{B}\left({\mathsf{\Omega }}_{\mathcal{P}}\right),m_{\mathcal{P}})$.

Proof. 

Fix ${\sigma }_0>\widehat{\sigma }$, and consider

$$a_m\left(\omega \right)=\frac{\omega \left(m\right)}{m^{{\sigma }_0}},m\in {\mathcal{N}}_{\mathcal{P}}.$$

Then $\{a_m:m\in {\mathcal{N}}_{\mathcal{P}}\}$ is a sequence of complex-valued random variables on $({\mathsf{\Omega }}_{\mathcal{P}},\mathcal{B}\left({\mathsf{\Omega }}_{\mathcal{P}}\right),m_{\mathcal{P}})$. Denote by $\overline{z}$ the complex conjugate of $z\in \mathbb{C}$. Suppose that $m_1\ne m_2$, $m_1,m_2\in {\mathcal{N}}_{\mathcal{P}}$. Since the set $L\left(\mathcal{P}\right)$ is linearly independent over $\mathbb{Q}$, in the product $\omega \left(m_1\right)\overline{\omega \left(m_2\right)}$, there exists at least one factor ${\omega }^{\alpha }\left(p\right)$, $p\in \mathcal{P}$, with integer $\alpha \ne 0$. Therefore, denoting by $\mathbb{E}\xi$ the expectation of the random variable $\xi$, we have

$$\mathbb{E}|a_m{\left(\omega \right)|}^2=\frac{1}{m^{2{\sigma }_0}},m\in {\mathcal{N}}_{\mathcal{P}},$$

(7)

$$\mathbb{E}{a}_{m_1}\left(\omega \right)\overline{a_{m_2}\left(\omega \right)}=\frac{1}{m_1^{{\sigma }_0}{m}_2^{{\sigma }_0}}\underset{{\mathsf{\Omega }}_{\mathcal{P}}}{\int }\omega \left(m_1\right)\overline{\omega \left(m_2\right)}\mathrm{d}{m}_{\mathcal{P}}=0,m_1\ne m_2,$$

because the integral includes the factor

$$\underset{\gamma }{\int }{\omega }^{\alpha }\left(p\right)\mathrm{d}{m}_{\gamma }=\underset{0}{\overset{1}{\int }}{\mathrm{e}}^{2\pi i\alpha u}\mathrm{d}u=0,$$

where $\gamma$ is the unit circle on $\mathbb{C}$, and $m_{\gamma }$ the Haar measure on $\gamma$. This and (7) show that $\left\{a_m\right\}$ is a sequence of pairwise orthogonal complex-valued random variables and the series

$$\sum _{m\in {\mathcal{N}}_{\mathcal{P}}}\mathbb{E}{\left|a_m\right|}^2{log}^{2}m$$

is convergent. Hence, by the classical Rademacher theorem, see [26], the series

$$\sum _{m\in {\mathcal{N}}_{\mathcal{P}}}\frac{\omega \left(m\right)}{m^{{\sigma }_0}}$$

converges for almost all $\omega$ with respect to the measure $m_{\mathcal{P}}$. Therefore, by a property of the Dirichlet series, see [22], the series

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

(8)

converges uniformly on compact sets of the half-plane $\sigma >{\sigma }_0$ for almost all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$.

Now, let

$${\sigma }_k=\widehat{\sigma }+\frac{1}{k},k\in \mathbb{N},$$

and $D_k=\{s\in \mathbb{C}:\sigma >{\sigma }_k\}$. Denote by the set ${\mathsf{\Omega }}_k\subset {\mathsf{\Omega }}_{\mathcal{P}}$ such that the series (8) converges uniformly on compact sets of $D_k$ for almost all $\omega \in {\mathsf{\Omega }}_k$. Then, by the above remark,

$$m_{\mathcal{P}}\left({\mathsf{\Omega }}_k\right)=1.$$

(9)

On the other hand, taking

$$\widehat{\mathsf{\Omega }}=\underset{k}{\cap }{\mathsf{\Omega }}_k,$$

we obtain from (9) that $m_{\mathcal{P}}\left(\widehat{\mathsf{\Omega }}\right)=1$, and the series (8) converges uniformly on compact sets of the half-plane $\sigma >\widehat{\sigma }$ of the strip D. Hence, ${\zeta }_{\mathcal{P}}(s,\omega )$ is the $H\left(D\right)$-valued random element on $({\mathsf{\Omega }}_{\mathcal{P}},\mathcal{B}\left({\mathsf{\Omega }}_{\mathcal{P}}\right),m_{\mathcal{P}})$. □

Lemma 2.

For almost all ω, the product

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

converges uniformly on compact subsets of the half-plane $\sigma >\widehat{\sigma }$, and the equality

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

holds.

Proof. 

The series ${\zeta }_{\mathcal{P}}(s,\omega )$ is absolutely convergent for $\sigma >1$. Therefore, the equality of the lemma, in view of (6), is valid for $\sigma >1$. By proof of Lemma 1, the function ${\zeta }_{\mathcal{P}}(s,\omega )$, for almost all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$, is analytic in the half-plane $\sigma >\widehat{\sigma }$. Therefore, by analytic continuation, it suffices to show that the product of the lemma, for almost all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$, converges uniformly on compact subsets of the strip D.

Write

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

(10)

with

$$a_p(s,\omega )=\sum _{\alpha =1}^{\infty }\frac{{\omega }^{\alpha }\left(p\right)}{p^{\alpha s}}.$$

We observe that the convergence of product (10) follows from that of the series

$$\sum _{p\in \mathcal{P}}{a}_p(s,\omega )\mathrm{and}\sum _{p\in \mathcal{P}}{\left|a_p(s,\omega )\right|}^2.$$

Set

$$b_p(s,\omega )=\frac{\omega \left(p\right)}{p^s}.$$

Then

$$a_p(s,\omega )-b_p(s,\omega )=\sum _{\alpha =2}^{\infty }\frac{{\omega }^{\alpha }\left(p\right)}{p^{\alpha s}}\ll \frac{1}{p^{2\sigma }},\sigma >\widehat{\sigma }.$$

Hence, the series

$$\sum _{p\in \mathcal{P}}|a_p(s,\omega )-b_p(s,\omega )|$$

(11)

is convergent for all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$ with every $\sigma ={\sigma }_0$, ${\sigma }_0>\widehat{\sigma }$, thus, uniformly convergent on compact subsets of the half-plane $\sigma >\widehat{\sigma }$. To prove the convergence for the series

$$\sum _{p\in \mathcal{P}}{b}_p(s,\omega ),$$

we apply the same arguments as in the proof of Lemma 1. For fixed $\sigma >\widehat{\sigma }$, we have

$$\mathbb{E}|b_p{(\sigma ,\omega )|}^2=\frac{1}{p^{2\sigma }}$$

and for $p,q\in \mathcal{P}$, $p\ne q$,

$$\mathbb{E}{b}_p(\sigma ,\omega )\overline{b_q(\sigma ,\omega )}=\frac{1}{p^{\sigma }{q}^{\sigma }}\underset{{\mathsf{\Omega }}_{\mathcal{P}}}{\int }\omega \left(p\right)\overline{\omega \left(q\right)}\mathrm{d}{m}_{\mathcal{P}}=0.$$

Thus, the series

$$\sum _{p\in \mathcal{P}}\mathbb{E}{\left|b_p(\sigma ,\omega )\right|}^2{log}^{2}p$$

is convergent, and the Rademacher theorem implies that the series

$$\sum _{p\in \mathcal{P}}{b}_p(\sigma ,\omega )$$

converges for almost all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$. Hence, this series, for almost all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$, converges uniformly on compact subsets of the half-plane $\sigma >\widehat{\sigma }$. This, together with a convergence property of the series (11), shows that the series

$$\sum _{p\in \mathcal{P}}{a}_p(s,\omega ),$$

for almost all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$, converges uniformly on compact subsets of the half-plane $\sigma >\widehat{\sigma }$, and it remains to prove the same for the series

$$\sum _{p\in \mathcal{P}}{\left|a_p(s,\omega )\right|}^2.$$

(12)

Clearly, for all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$,

$$|a_p{(s,\omega )|}^2\ll \frac{1}{p^{2\sigma }},\sigma >\widehat{\sigma }.$$

Hence, the series (12), for all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$, converges uniformly on compact subsets of the half-plane $\sigma >\widehat{\sigma }$. □

3. Ergodicity

For $\tau \in \mathbb{R}$, let

$${\kappa }_{\tau }=\left(p^{-i\tau }:p\in \mathcal{P}\right),$$

and

$$g_{\tau }\left(\omega \right)={\kappa }_{\tau }\omega ,\omega \in {\mathsf{\Omega }}_{\mathcal{P}}.$$

Since the Haar measure $m_{\mathcal{P}}$ is invariant with respect to shifts by elements of ${\mathsf{\Omega }}_{\mathcal{P}}$, i.e., for all $A\in \mathcal{B}\left({\mathsf{\Omega }}_{\mathcal{P}}\right)$ and $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$,

$$m_{\mathcal{P}}\left(A\right)=m_{\mathcal{P}}\left(\omega A\right)=m_{\mathcal{P}}\left(A\omega \right),$$

$g_{\tau }\left(m\right)$ is a measurable measure preserving transformation on ${\mathsf{\Omega }}_{\mathcal{P}}$. Thus, we have the one-parameter group $G_{\tau }=\{g_{\tau }:\tau \in \mathbb{R}\}$ of transformations of ${\mathsf{\Omega }}_{\mathcal{P}}$. A set $A\in \mathcal{B}\left({\mathsf{\Omega }}_{\mathcal{P}}\right)$ is called invariant with respect to $G_{\tau }$ if, for every $\tau \in \mathbb{R}$, the sets A and $A_{\tau }=g_{\tau }\left(A\right)$ differ one from another at most by a set of $m_{\mathcal{P}}$-measure zero. It is well known that all invariant sets form a $\sigma$-field which is a subfield of $\mathcal{B}\left({\mathsf{\Omega }}_{\mathcal{P}}\right)$. The group $G_{\tau }$ is called ergodic if its $\sigma$-field of invariant sets consists only of sets $m_{\mathcal{P}}$-measure 0 or 1.

Lemma 3.

Under the hypotheses of Theorem 2, the group $G_{\tau }$ is ergodic.

Proof. 

Let $A\in \mathcal{B}\left({\mathsf{\Omega }}_{\mathcal{P}}\right)$ be a fixed invariant set of $G_{\tau }$. Denote by ${\mathrm{I}}_A\left(\omega \right)$ the indicator function of the set A. Then, for almost all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$,

$${\mathrm{I}}_A\left(g_{\tau }\left(\omega \right)\right)={\mathrm{I}}_A\left(\omega \right).$$

(13)

Characters $\chi$ of the group ${\mathsf{\Omega }}_{\mathcal{P}}$ are of the form

$$\chi \left(\omega \right)=\underset{p\in \mathcal{P}}{{\prod }^{*}}{\omega }^{k_p}\left(p\right),$$

(14)

where * indicates that only a finite number of integers $k_p$ are distinct from zero. Suppose that $\chi$ is a nontrivial character, i.e., $\chi \left(\omega \right)\neg \equiv 1$ for all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$. Then, we have

$$\chi \left(g_{\tau }\right)=\underset{p\in \mathcal{P}}{{\prod }^{*}}{p}^{-i{k}_p\tau }=exp\left\{-i\tau \underset{p\in \mathcal{P}}{{\sum }^{*}}{k}_{p}logp\right\}.$$

Since the set $L\left(\mathcal{P}\right)$ is linearly independent over $\mathbb{Q}$, and $\chi$ is a nontrivial character,

$$\underset{p\in \mathcal{P}}{{\sum }^{*}}{k}_{p}logp\ne 0.$$

Thus, there exists a real number $a\ne 0$ such that

$$\chi \left(g_{\tau }\right)={\mathrm{e}}^{-i\tau a}.$$

Hence, there is ${\tau }_0\in \mathbb{R}$ satisfying $\chi \left(g_{{\tau }_0}\right)\ne 1$.

Now, we deal with Fourier analysis on ${\mathsf{\Omega }}_{\mathcal{P}}$. Denote by $\widehat{g}$ the Fourier transform of a function g, i.e.,

$$\widehat{g}\left(\chi \right)=\underset{{\mathsf{\Omega }}_{\mathcal{P}}}{\int }g\left(\omega \right)\chi \left(\omega \right)\mathrm{d}{m}_{\mathcal{P}}.$$

In virtue of (13), we find

$${\widehat{\mathrm{I}}}_A\left(\chi \right)=\underset{{\mathsf{\Omega }}_{\mathcal{P}}}{\int }{\mathrm{I}}_A\left(\omega \right)\chi \left(\omega \right)\mathrm{d}{m}_{\mathcal{P}}=\chi \left(g_{{\tau }_0}\right)\underset{{\mathsf{\Omega }}_{\mathcal{P}}}{\int }\chi \left(\omega \right){\mathrm{I}}_A\left(\omega \right)\mathrm{d}{m}_{\mathcal{P}}=\chi \left(g_{{\tau }_0}\right){\widehat{\mathrm{I}}}_A\left(\chi \right).$$

Hence, in view of inequality $\chi \left(g_{{\tau }_0}\right)\ne 1$, we obtain

$${\widehat{\mathrm{I}}}_A\left(\chi \right)=0.$$

(15)

Consider the case of the trivial character ${\chi }_0$ of the group ${\mathsf{\Omega }}_{\mathcal{P}}$. We set ${\widehat{\mathrm{I}}}_A\left({\chi }_0\right)=c$. Then, the orthogonality of characters implies that

$$\widehat{c}\left(\chi \right)=\underset{{\mathsf{\Omega }}_{\mathcal{P}}}{\int }c\left(\chi \right)\chi \left(\omega \right)\mathrm{d}{m}_{\mathcal{P}}=c\underset{{\mathsf{\Omega }}_{\mathcal{P}}}{\int }\chi \left(\omega \right)\mathrm{d}{m}_{\mathcal{P}}=\left\{\begin{array}{cc}c\hfill & \mathrm{if}\chi ={\chi }_0,\hfill \\ 0\hfill & \mathrm{if}\chi \ne {\chi }_0.\hfill \end{array}\right.$$

Therefore, using (15) yields the equality

$${\widehat{\mathrm{I}}}_A\left(\chi \right)=\widehat{c}\left(\chi \right).$$

(16)

It is well known that a function is completely determined by its Fourier transform. Thus, by (16), we have that for almost all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$, ${\mathrm{I}}_A\left(\omega \right)=c$. However, as ${\mathrm{I}}_A\left(\omega \right)$ is the indicator function, it follows that $c=0$ or 1. In other words, for almost all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$, ${\mathrm{I}}_A\left(\omega \right)=0$ or ${\mathrm{I}}_A\left(\omega \right)=1$. Thus, $m_{\mathcal{P}}\left(A\right)=0$ or $m_{\mathcal{P}}\left(A\right)=1$. The lemma is proved. □

We apply Lemma 3 for the estimation of the mean square for ${\zeta }_{\mathcal{P}}(s,\omega )$.

Lemma 4.

Under hypotheses of Theorem 2, for fixed $\widehat{\sigma }<\sigma <1$ and almost all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$,

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

Proof. 

Let $a_m(\sigma ,\omega )$, $m\in {\mathcal{N}}_{\mathcal{P}}$, be the same as the proof of Lemma 1. The random variables $a_m(\sigma ,\omega )$ are pairwise orthogonal, and

$$\mathbb{E}{\left|a_m(\sigma ,\omega )\right|}^2=\frac{1}{m^{2\sigma }}.$$

Therefore,

$$\mathbb{E}{\left|{\zeta }_{\mathcal{P}}(\sigma ,\omega )\right|}^2=\mathbb{E}{\left|\sum _{m\in {\mathcal{N}}_{\mathcal{P}}}{a}_m(\sigma ,\omega )\right|}^2=\sum _{m\in {\mathcal{N}}_{\mathcal{P}}}\mathbb{E}{\left|a_m(\sigma ,\omega )\right|}^2=\sum _{m\in {\mathcal{N}}_{\mathcal{P}}}\frac{1}{m^{2\sigma }}<\infty .$$

(17)

Let $g_{\tau }\left(\omega \right)$ be the transformation from the proof of Lemma 3. Then, by the definition of $g_{\tau }$,

$${\left|{\zeta }_{\mathcal{P}}(\sigma ,g_t\left(\omega \right)\right|}^2={\left|{\zeta }_{\mathcal{P}}(\sigma ,g_t\omega )\right|}^2={\left|{\zeta }_{\mathcal{P}}(\sigma +it,\omega )\right|}^2.$$

We recall that a strongly stationary random process $X(t,\omega )$, $t\in \mathcal{T}$, on $(\mathsf{\Omega },\mathcal{A},P)$ is called ergodic if its $\sigma$-field of invariant sets consists of sets of P-measure 0 or 1. Since the group $G_{\tau }$ is ergodic, the stationary process $|{\zeta }_{\mathcal{P}}{(\sigma +it,\omega )|}^2$ is ergodic, for details, see [22]. Therefore, we can apply the classical Birkhoff–Khintchine ergodic theorem, see [27]. This gives, by (17),

$$\underset{T\to \infty }{lim}\frac{1}{2T}\underset{-T}{\overset{T}{\int }}{\left|{\zeta }_{\mathcal{P}}(\sigma +it,\omega )\right|}^2\mathrm{d}t=\underset{T\to \infty }{lim}\frac{1}{2T}\underset{-T}{\overset{T}{\int }}{\zeta }_{\mathcal{P}}(\sigma ,g_t\left(\omega \right))\mathrm{d}t=\mathbb{E}{\left|{\zeta }_{\mathcal{P}}(\sigma ,\omega )\right|}^2<\infty .$$

□

4. Approximation in the Mean

In this section, we approximate the functions ${\zeta }_{\mathcal{P}}\left(s\right)$ and ${\zeta }_{\mathcal{P}}(s,\omega )$ by absolutely convergent Dirichlet series. Let $\eta >1-\widehat{\sigma }$ be a fixed number, and, for $m\in {\mathcal{N}}_{\mathcal{P}}$ and $n\in \mathbb{N}$,

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

Then the series

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

are absolutely convergent for $\sigma >\widehat{\sigma }$ and for every fixed $n\in \mathbb{N}$. We will approximate ${\zeta }_{\mathcal{P}}\left(s\right)$ and ${\zeta }_{\mathcal{P}}(s,\omega )$ by ${\zeta }_{\mathcal{P},n}\left(s\right)$ and ${\zeta }_{\mathcal{P},n}(s,\omega )$, respectively, in the mean. Recall a metric in the space $H\left(D\right)$ inducing its topology. Let $\{K_l:l\in \mathbb{N}\}\subset D$ be a sequence of embedded compact sets such that

$$D=\underset{l=1}{\overset{\infty }{\cup }}{K}_l,$$

and every compact set $K\subset D$ lies in some $K_l$. Then

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

is the desired metric in $H\left(D\right)$.

In [24], the following statement has been obtained.

Lemma 5.

Suppose that (1) is valid. Then

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

Denote by ${\mathsf{\Omega }}_{\mathcal{P},1}$ a subset of ${\mathsf{\Omega }}_{\mathcal{P}}$ such that a product

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

converges uniformly on compact subsets of D for $\omega \in {\mathsf{\Omega }}_{\mathcal{P},1}$, and by ${\mathsf{\Omega }}_{\mathcal{P},2}$ a subset of ${\mathsf{\Omega }}_{\mathcal{P}}$ such that, for $\omega \in {\mathsf{\Omega }}_{\mathcal{P},2}$, the estimate

$$\underset{-T}{\overset{T}{\int }}{\left|{\zeta }_{\mathcal{P}}(\sigma +it,\omega )\right|}^2\mathrm{d}t{\ll }_{\sigma }T$$

holds for $\widehat{\sigma }<\sigma <1$. Then, by Lemmas 3 and 4, $m_{\mathcal{P}}\left({\mathsf{\Omega }}_{\mathcal{P},j}\right)=1$, $j=1,2$. Let

$${\widehat{\mathsf{\Omega }}}_{\mathcal{P}}={\mathsf{\Omega }}_{\mathcal{P},1}\cap {\mathsf{\Omega }}_{\mathcal{P},2}.$$

Then again $m_{\mathcal{P}}\left({\widehat{\mathsf{\Omega }}}_{\mathcal{P}}\right)=1$.

Lemma 6.

Under the hypotheses of Theorem 2, for $\omega \in {\widehat{\mathsf{\Omega }}}_{\mathcal{P}}$ the equality

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

holds.

Proof. 

Denote

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

where $\mathsf{\Gamma }\left(s\right)$ is the Euler gamma function. Then the classical Mellin formula

$$\frac{1}{2\pi i}\underset{a-i\infty }{\overset{a+i\infty }{\int }}\mathsf{\Gamma }\left(z\right)b^{-z}\mathrm{d}z={\mathrm{e}}^{-b},a,b>0,$$

implies, for $m\in {\mathcal{N}}_{\mathcal{P}}$,

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

Therefore, for $\sigma >\widehat{\sigma }$ and $\omega \in {\widehat{\mathsf{\Omega }}}_{\mathcal{P}}$,

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

(18)

The definition of the metric $\rho$ implies that it is sufficient to show that, for every compact set $K\subset D$,

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

(19)

Thus, let $K\subset D$ be a compact set. Then there exists $\epsilon >0$ satisfying for $\sigma +it\in K$ the inequalities $\widehat{\sigma }+\epsilon ⩽\sigma ⩽1-\epsilon /2$. Take $\eta =1$ and ${\eta }_1=\widehat{\sigma }-\epsilon /2-\sigma$ with the above $\sigma$. Then ${\eta }_1<0$ and ${\eta }_1⩾\widehat{\sigma }+\epsilon /2-1+\epsilon /2=\widehat{\sigma }-1+\epsilon >-1$. Consequently, the integrand in (18) has only a simple pole $z=0$ in the strip ${\eta }_1<\mathrm{Re}z<\eta$. Hence, the residue theorem and (18) show that, for $s\in K$,

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

Thus, for $s\in K$,

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

(20)

It is well known that, for the gamma-function $\mathsf{\Gamma }(\sigma +it)$, the estimate

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

(21)

is valid uniformly for $\sigma \in [{\sigma }_1,{\sigma }_2]$ with every ${\sigma }_1<{\sigma }_2$. Therefore, (20) implies

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

(22)

By Lemma 4, for $\omega \in {\widehat{\mathsf{\Omega }}}_{\mathcal{P}}$,

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

Hence,

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

(23)

In view of (21), for $s\in K$,

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

This and (23) give

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

and (19) is proved. □

5. Limit Theorems

In previous sections, we gave preparatory results for the proof of a limit theorem for ${\zeta }_{\mathcal{P}}\left(s\right)$ in the space of analytic functions $H\left(D\right)$. In this section, we consider the weak convergence for

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

and

$${\widehat{P}}_{T,\mathcal{P}}\left(A\right)=\frac{1}{T}\mathrm{meas}\left\{\tau \in [0,T]:{\zeta }_{\mathcal{P}}(s+i\tau ,\omega )\in A\right\}$$

as $T\to \infty$, where $A\in \mathcal{B}\left(H\right(D\left)\right)$, $\omega \in {\widehat{\mathsf{\Omega }}}_{\mathcal{P}}$.

We start with a limit lemma on ${\mathsf{\Omega }}_{\mathcal{P}}$. For $A\in \mathcal{B}\left({\mathsf{\Omega }}_{\mathcal{P}}\right)$, define

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

Lemma 7.

Suppose that the set $L\left(\mathcal{P}\right)$ is linearly independent over $\mathbb{Q}$. Then $P_{T,\mathcal{P}}^{{\mathsf{\Omega }}_{\mathcal{P}}}$ converges weakly to the Haar measure $m_{\mathcal{P}}$ as $T\to \infty$.

Proof. 

In the proof of Lemma 3, we have seen that characters of the group ${\mathsf{\Omega }}_{\mathcal{P}}$ are given by (14). Therefore, the Fourier transform $F_{T,\mathcal{P}}\left(\underline{k}\right)$, $\underline{k}=(k_p:k_p\in \mathbb{Z},p\in \mathcal{P})$ of $P_{T,\mathcal{P}}^{{\mathsf{\Omega }}_{\mathcal{P}}}$ is defined by

$$\begin{array}{cc}\hfill F_{T,\mathcal{P}}\left(\underline{k}\right)& =\underset{{\mathsf{\Omega }}_{\mathcal{P}}}{\int }\underset{p\in \mathcal{P}}{{\prod }^{*}}{\omega }^{k_p}\left(p\right)\mathrm{d}{P}_{T,\mathcal{P}}^{{\mathsf{\Omega }}_{\mathcal{P}}}=\frac{1}{T}\underset{0}{\overset{T}{\int }}\left(\underset{p\in \mathcal{P}}{{\prod }^{*}}{p}^{-i\tau k_p}\right)\mathrm{d}\tau \hfill \\ & =\frac{1}{T}\underset{0}{\overset{T}{\int }}exp\left\{-i\tau \underset{p\in \mathcal{P}}{{\sum }^{*}}{k}_{p}logp\right\}\mathrm{d}\tau .\hfill \end{array}$$

(24)

We have to show that

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

(25)

For this, we apply the linear independence of the set $L\left(\mathcal{P}\right)$. We have

$$A_{\mathcal{P}}\left(\underline{k}\right)\stackrel{\mathrm{def}}{=}\underset{p\in \mathcal{P}}{{\sum }^{*}}{k}_{p}logp=0$$

if and only if ${\underline{k}}_p=\underline{0}$. Thus, (24),

$$F_{T,\mathcal{P}}\left(\underline{k}\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\underline{k}=\underline{0},\hfill \\ \frac{1-exp\{-iT{A}_{\mathcal{P}}\left(\underline{k}\right)\}}{iTexp\{-i{A}_{\mathcal{P}}\left(\underline{k}\right)\}}\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

and (25) take place. □

The next lemma is devoted to the functions ${\zeta }_{\mathcal{P},n}\left(s\right)$ and ${\zeta }_{\mathcal{P},n}(s,\omega )$. For $A\in \mathcal{B}\left(H\right(D\left)\right)$, set

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

and

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

Lemma 8.

Suppose that the set $L\left(\mathcal{P}\right)$ is linearly independent over $\mathbb{Q}$. Then, on $\left(H\right(D),\mathcal{B}(H\left(D\right)\left)\right)$ there exists a probability measure $P_{\mathcal{P},n}$ such that both the measures $P_{T,\mathcal{P},n}$ and ${\widehat{P}}_{T,\mathcal{P},n}$ converge weakly to $P_{\mathcal{P},n}$ as $T\to \infty$.

Proof. 

We use a property of the preservation of weak convergence under continuous mappings. Consider the mapping $v_{\mathcal{P},n}:{\mathsf{\Omega }}_{\mathcal{P}}\to H\left(D\right)$ given by

$$v_{\mathcal{P},n}\left(\omega \right)={\zeta }_{\mathcal{P},n}(s,\omega ).$$

Since the series for ${\zeta }_{\mathcal{P},n}(s,\omega )$ is absolutely convergent for $\sigma >\widehat{\sigma }$, the mapping $v_{\mathcal{P},n}$ is continuous. Moreover, for $A\in \mathcal{B}\left(H\right(D\left)\right)$,

$$P_{T,\mathcal{P},n}\left(A\right)=\frac{1}{T}\mathrm{meas}\left\{\tau \in [0,T]:\left(p^{-i\tau }:p\in \mathcal{P}\right)\in v_{\mathcal{P},n}^{-1}A\right\}=P_{T,\mathcal{P}}^{{\mathsf{\Omega }}_{\mathcal{P}}}\left(v_{\mathcal{P},n}^{-1}A\right).$$

Thus, denoting by $P_{T,\mathcal{P}}^{{\mathsf{\Omega }}_{\mathcal{P}}}{v}_{\mathcal{P},n}^{-1}$ the measure given by the latter equality, we obtain that $P_{T,\mathcal{P},n}=P_{T,\mathcal{P}}^{{\mathsf{\Omega }}_{\mathcal{P}}}{v}_{\mathcal{P},n}^{-1}$. This equality continuity of $v_{\mathcal{P},n}$, and the principle of preservation of weak convergence, see Theorem 5.1 of [28], show that $P_{T,\mathcal{P},n}$ converges weakly to the measure $Q_{\mathcal{P},n}\stackrel{\mathrm{def}}{=}{m}_{\mathcal{P}}{v}_{\mathcal{P},n}^{-1}$ as $T\to \infty$.

Define one more mapping ${\widehat{v}}_{\mathcal{P},n}:{\mathsf{\Omega }}_{\mathcal{P}}\to H\left(D\right)$ by

$${\widehat{v}}_{\mathcal{P},n}\left(\widehat{\omega }\right)={\zeta }_{\mathcal{P},n}(s,\omega \widehat{\omega }),\widehat{\omega }\in {\mathsf{\Omega }}_{\mathcal{P}}.$$

Then, repeating the above arguments, we find that ${\widehat{P}}_{T,\mathcal{P},n}$ converges weakly to ${\widehat{Q}}_{\mathcal{P},n}\stackrel{\mathrm{def}}{=}{m}_{\mathcal{P}}{\widehat{v}}_{\mathcal{P},n}^{-1}$. Let $v_{\mathcal{P}}\left(\widehat{\omega }\right)=\omega \widehat{\omega }$. Then, by invariance of the measure $m_{\mathcal{P}}$, we have

$${\widehat{Q}}_{\mathcal{P},n}=m_{\mathcal{P}}{\left(v_{\mathcal{P},n}{v}_{\mathcal{P}}\right)}^{-1}=\left(m_{\mathcal{P}}{v}_{\mathcal{P}}^{-1}\right)v_{\mathcal{P},n}^{-1}=m_{\mathcal{P}}{v}_{\mathcal{P},n}^{-1}=Q_{\mathcal{P},n}.$$

Thus, $P_{T,\mathcal{P},n}$ and ${\widehat{P}}_{T,\mathcal{P},n}$ converge weakly to the same measure $Q_{\mathcal{P},n}$ as $T\to \infty$. □

Next, we study the family of probability measures $\{Q_{\mathcal{P},n}:n\in \mathbb{N}\}$. We recall some notions. A family of probability measures $\left\{P\right\}$ on $(\mathbb{X},\mathcal{B}(\mathbb{X}\left)\right)$ is called tight if, for every $\epsilon >0$, there exists a compact set $K\subset \mathbb{X}$ such that

$$P\left(K\right)>1-\epsilon$$

for all P, and $\left\{P\right\}$ is relatively compact if every sequence $\left\{P_k\right\}\subset \left\{P\right\}$ has a subsequence $\left\{P_{n_k}\right\}$ weakly convergent to a certain probability measure P on $(\mathbb{X},\mathcal{B}(\mathbb{X}\left)\right)$ as $k\to \infty$. By the classical Prokhorov theorem, see Theorem 6.1 of [28], every tight family of probability measures is relatively compact.

Lemma 9.

Under the hypotheses of Theorem 2, the family $\{Q_{\mathcal{P},n}:n\in \mathbb{N}\}$ is relatively compact.

Proof. 

In view of the above remark, it suffices to prove the tightness of $\left\{Q_{\mathcal{P},n}\right\}$. Let $K\subset D$ be a compact. Then, using the Cauchy integral formula and absolute convergence of the series for ${\zeta }_{\mathcal{P},n}\left(s\right)$, we obtain ${\sigma }_{\kappa }>\widehat{\sigma }$

$$\underset{n\in \mathbb{N}}{sup}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}{\left|{\zeta }_{\mathcal{P},n}(s+i\tau )\right|}^2\mathrm{d}\tau \ll \underset{n\in \mathbb{N}}{sup}\sum _{m\in {\mathcal{N}}_{\mathcal{P}}}\frac{a_n^2\left(m\right)}{m^{2{\sigma }_{\kappa }}}\ll \sum _{m\in {\mathcal{N}}_{\mathcal{P}}}\frac{1}{m^{2{\sigma }_{\kappa }}}\stackrel{\mathrm{def}}{=}{V}_{\kappa }<\infty .$$

(26)

Suppose that ${\xi }_T$ is a random variable on a certain probability space $(\mathsf{\Xi },\mathcal{A},\mu )$ uniformly distributed in the interval $[0,T]$. Define the $H\left(D\right)$-valued random element

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

Then, denoting by $\stackrel{\mathcal{D}}{\to }$ the convergence in distribution by Lemma 8, we obtain

$$Y_{T,\mathcal{P},n}\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{Y}_{\mathcal{P},n},$$

(27)

where $Y_{\mathcal{P},n}\left(s\right)$ is the $H\left(D\right)$-valued random element with the distribution $Q_{\mathcal{P},n}$. Since the convergence in $H\left(D\right)$ is uniform on compact sets, (27) implies

$$\underset{s\in K}{sup}\left|Y_{T,\mathcal{P},n}\left(s\right)\right|\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}\underset{s\in K}{sup}\left|Y_{\mathcal{P},n}\left(s\right)\right|.$$

(28)

Now, let $K=K_l$, where $\left\{K_l\right\}$ is a sequence of compact sets of D from the definition of the metric $\rho$. Fix $\epsilon >0$, and set $R_l=2^l{\epsilon }^{-1}\sqrt{V_l}$ where $V_l=V_{{\kappa }_l}$. Therefore, relation (26), and the Chebyshev type inequality yield

$$\begin{array}{cc}\hfill \underset{T\to \infty }{lim\; sup}\mu \left\{\underset{s\in K_l}{sup}\left|Y_{T,\mathcal{P},n}\left(s\right)\right|>R_l\right\}& ⩽\underset{n\in \mathbb{N}}{sup}\underset{T\to \infty }{lim\; sup}\frac{1}{T{R}_l}\underset{0}{\overset{T}{\int }}\underset{s\in K_l}{sup}\left|{\zeta }_{\mathcal{P},n}(s+i\tau )\right|\mathrm{d}\tau \hfill \\ & ⩽\underset{s\in \mathbb{N}}{sup}\underset{T\to \infty }{lim\; sup}\frac{1}{R_l}{\left(\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K_l}{sup}{\left|{\zeta }_{\mathcal{P},n}(s+i\tau )\right|}^2\mathrm{d}\tau \right)}^{1/2}=\frac{\epsilon }{2^l}.\hfill \end{array}$$

Hence, in view of (28),

$$\mu \left\{\underset{s\in K_l}{sup}\left|Y_{\mathcal{P},n}\left(s\right)\right|>R_l\right\}⩽\frac{\epsilon }{2^l}.$$

(29)

Define the set

$$H\left(\epsilon \right)=\left\{g\in H\left(D\right):\underset{s\in K_l}{sup}\left|g\left(s\right)\right|⩽R_l,l\in \mathbb{N}\right\}.$$

Then $H\left(\epsilon \right)$ is a compact set in $H\left(D\right)$. Moreover, inequality (29) implies that

$$\mu \left\{Y_{\mathcal{P},n}\in H\left(\epsilon \right)\right\}=1-\mu \left\{Y_{\mathcal{P},n}\notin H\left(\epsilon \right)\right\}⩾1-\epsilon \sum _{l=1}^{\infty }\frac{1}{2^l}=1-\epsilon$$

for all $n\in \mathbb{N}$. Since $Q_{\mathcal{P},n}$ is the distribution of $Y_{\mathcal{P},n}$, this shows that

$$Q_{\mathcal{P},n}\left(H\left(\epsilon \right)\right)⩾1-\epsilon$$

for all $n\in \mathbb{N}$. The lemma is proved. □

Now, we are ready to consider the weak convergence for $P_{T,\mathcal{P}}$ and ${\widehat{P}}_{T,\mathcal{P}}$. For convenience, we recall one general statement.

Proposition 1.

Suppose that a metric space $(\mathbb{X},d)$ is separable, and the $\mathbb{X}$-valued random elements $x_{mn}$ and $y_n$, $m,n\in \mathbb{N}$ are defined on the same probability space $(\mathsf{\Xi },\mathcal{A},\mu )$. Suppose that

$$x_{mn}\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}{x}_m,x_m\underset{m\to \infty }{\overset{\mathcal{D}}{\to }}x,$$

and, for every $\epsilon >0$,

$$\underset{m\to \infty }{lim}\underset{n\to \infty }{lim\; sup}\mu \left\{d(x_{mn},y_n)⩾\epsilon \right\}=0$$

Then

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

Proof. 

The proposition is Theorem 4.2 of [28], where its proof is given. □

Lemma 10.

Under the hypotheses of Theorem 2, on $\left(H\right(D),\mathcal{B}(H\left(D\right)\left)\right)$ there exists a probability measure $P_{\mathcal{P}}$ such that both the measures $P_{T,\mathcal{P}}$ and ${\widehat{P}}_{T,\mathcal{P}}$ converge weakly to $P_{\mathcal{P}}$ as $T\to \infty$.

Proof. 

Let ${\xi }_T$ be the same random variable as in the proof of Lemma 9. By Lemma 9, there exists a sequence $\left\{Q_{\mathcal{P},n_m}\right\}\subset \left\{Q_{\mathcal{P},n}\right\}$ and the probability measure $Q_{\mathcal{P}}$ on $\left(H\right(D),\mathcal{B}(H\left(D\right)\left)\right)$ such that $Q_{\mathcal{P},n_m}$ converges weakly to $Q_{\mathcal{P}}$ as $m\to \infty$. In other words, in the notation of the proof of Lemma 9,

$$Y_{\mathcal{P},n_m}\underset{m\to \infty }{\overset{\mathcal{D}}{\to }}{Q}_{\mathcal{P}}.$$

(30)

On $(\mathsf{\Xi },\mathcal{A},\mu )$, define one more $H\left(D\right)$-valued random element

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

Then the application of Lemma 5 gives, for $\epsilon >0$,

$$\begin{array}{cc}\hfill \underset{m\to \infty }{lim}& \underset{T\to \infty }{lim\; sup}\mu \left\{\rho (Y_{T,\mathcal{P}},Y_{T,\mathcal{P},n_m})⩾\epsilon \right\}\hfill \\ & =\underset{m\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\mathrm{meas}\left\{\tau \in [0,T]:\rho \left({\zeta }_{\mathcal{P}}(s+i\tau ),{\zeta }_{\mathcal{P},n_m}(s+i\tau )\right)⩾\epsilon \right\}\hfill \\ & ⩽\underset{m\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{\epsilon T}\underset{0}{\overset{T}{\int }}\rho \left({\zeta }_{\mathcal{P}}(s+i\tau ),{\zeta }_{\mathcal{P},n_m}(s+i\tau )\right)\mathrm{d}\tau =0.\hfill \end{array}$$

This, and relations (27) and (30) show that all conditions of Proposition 1 are fulfilled. Thus, we have

$$Y_{T,\mathcal{P}}\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{Q}_{\mathcal{P}},$$

(31)

$P_{T,\mathcal{P}}$ converges weakly to $Q_{\mathcal{P}}$ as $T\to \infty$. Since the family $\left\{Q_{\mathcal{P},n}\right\}$ is relatively compact, relation (31), in addition, implies that

$$Y_{\mathcal{P},n}\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}{Q}_{\mathcal{P}}.$$

(32)

It remains to prove weak convergence for ${\widehat{P}}_{T,\mathcal{P}}$. On $(\mathsf{\Xi },\mathcal{A},\mu )$, define the $H\left(D\right)$-valued random elements

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

and

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

Lemma 8 implies the relation

$${\widehat{Y}}_{T,\mathcal{P},n}\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{Q}_{\mathcal{P}},$$

(33)

while, in view of Lemma 6, for $\epsilon >0$,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}& \underset{T\to \infty }{lim\; sup}\mu \left\{\rho \left({\widehat{Y}}_{T,\mathcal{P}},{\widehat{Y}}_{T,\mathcal{P},n}\right)⩾\epsilon \right\}\hfill \\ & ⩽\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{\epsilon T}\underset{0}{\overset{T}{\int }}\rho \left({\zeta }_{\mathcal{P}}(s+i\tau ,\omega ),{\zeta }_{\mathcal{P},n}(s+i\tau ,\omega )\right)\mathrm{d}\tau =0.\hfill \end{array}$$

This, (32), (33) and Lemma 10 yield the relation

$${\widehat{Y}}_{T,\mathcal{P}}\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{Q}_{\mathcal{P}}.$$

Thus, ${\widehat{P}}_{T,\mathcal{P}}$, as $T\to \infty$, also converges weakly to $Q_{\mathcal{P}}$. □

It remains to identify the measure $Q_{\mathcal{P}}$. Denote by $P_{{\zeta }_{\mathcal{P}}}$ the distribution of the random element ${\zeta }_{\mathcal{P}}(s,\omega )$, i.e.,

$$P_{{\zeta }_{\mathcal{P}}}\left(A\right)=m_{\mathcal{P}}\left\{\omega \in {\mathsf{\Omega }}_{\mathcal{P}}:{\zeta }_{\mathcal{P}}(s,\omega )\in A\right\}.$$

Theorem 3.

Under hypotheses of Theorem 2, $P_{T,\mathcal{P}}$ converges weakly to the measure $P_{{\zeta }_{\mathcal{P}}}$ as $T\to \infty$.

Proof. 

We will show that the limit measure $Q_{\mathcal{P}}$ in Lemma 10 coincides with $P_{{\zeta }_{\mathcal{P}}}$.

We apply the equivalent of weak convergence of probability measures in terms of continuity sets, see Theorem 2.1 of [28]. Let A be a continuity set of the measure $Q_{\mathcal{P}}$, i.e., $Q_{\mathcal{P}}(\partial A)=0$, where $\partial A$ denotes the boundary of A. Then, Lemma 10 implies that

$$\underset{T\to \infty }{lim}{\widehat{P}}_{T,\mathcal{P}}\left(A\right)=Q_{\mathcal{P}}\left(A\right).$$

(34)

On $({\mathsf{\Omega }}_{\mathcal{P}},\mathcal{B}\left({\mathsf{\Omega }}_{\mathcal{P}}\right))$, define the random variable

$${\xi }_{\mathcal{P}}\left(\omega \right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}{\zeta }_{\mathcal{P}}(s,\omega )\notin A,\hfill \\ 1\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Return to the group $G_{\tau }$ of Lemma 3. Since, by Lemma 3, the group $G_{\tau }$ is ergodic, the process $\xi \left(g_{\tau }\left(\omega \right)\right)$ is ergodic, and application of the Birkhoff–Khintchine theorem [27] gives

$$\underset{T\to \infty }{lim}\frac{1}{T}\underset{0}{\overset{T}{\int }}{\xi }_{\mathcal{P}}\left(g_{\tau }\left(\omega \right)\right)\mathrm{d}\tau =\mathbb{E}{\xi }_{\mathcal{P}}\left(\omega \right)$$

(35)

for almost all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$. However, the definition of the random variable ${\xi }_T\left(\omega \right)$ implies that, for almost all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$,

$$\begin{array}{cc}\hfill \frac{1}{T}\underset{0}{\overset{T}{\int }}{\xi }_{\mathcal{P}}\left(g_{\tau }\left(\omega \right)\right)\mathrm{d}\tau & =\frac{1}{T}\mathrm{meas}\left\{\tau \in [0,T]:{\zeta }_{\mathcal{P}}(s,g_{\tau }\left(\omega \right))\in A\right\}\hfill \\ & =\frac{1}{T}\mathrm{meas}\left\{\tau \in [0,T]:{\zeta }_{\mathcal{P}}(s+i\tau ,\omega )\in A\right\}.\hfill \end{array}$$

Thus, by (34),

$$\underset{T\to \infty }{lim}\frac{1}{T}\underset{0}{\overset{T}{\int }}{\xi }_{\mathcal{P}}\left(g_{\tau }\left(\omega \right)\right)\mathrm{d}\tau =Q_{\mathcal{P}}\left(A\right).$$

(36)

Moreover,

$$\mathbb{E}\xi \left(\omega \right)=\underset{{\mathsf{\Omega }}_{\mathcal{P}}}{\int }{\xi }_{\mathcal{P}}\left(\omega \right)\mathrm{d}{m}_{\mathcal{P}}=P_{{\zeta }_{\mathcal{P}}}\left(A\right).$$

This, (35) and (36) prove that $Q_{\mathcal{P}}\left(A\right)=P_{{\zeta }_{\mathcal{P}}}\left(A\right)$ for all continuity sets A of the measure $Q_{\mathcal{P}}$. It is well known that all continuity sets constitute a determining class. Hence, we have $Q_{\mathcal{P}}=P_{{\zeta }_{\mathcal{P}}}$, and the theorem is proved. □

6. Support

For the proof of Theorem 2, the explicitly given support of the measure $P_{{\zeta }_{\mathcal{P}}}$ is needed. We recall that the support of $P_{{\zeta }_{\mathcal{P}}}$ is a minimal closed set $S_{\mathcal{P}}\subset H\left(D\right)$ such that $P_{{\zeta }_{\mathcal{P}}}\left(S_{\mathcal{P}}\right)=1$. Every open neighbourhood of elements $S_{\mathcal{P}}$ has a positive $P_{{\zeta }_{\mathcal{P}}}$-measure.

Define the set

$$S_{\mathcal{P}}=\left\{g\in H\left(D\right):g\left(s\right)\ne 0\mathrm{or}g\left(s\right)\equiv 0\right\}.$$

Proposition 2.

Under the hypotheses of Theorem 2, the support of the measure $P_{{\zeta }_{\mathcal{P}}}$ is the set $S_{\mathcal{P}}$.

A proof of Proposition 2 is similar to that in the case of the Riemann zeta-function. Therefore, we will state without proof only the lemmas because their proofs word for word coincide with analogical assertions from [22].

We start with some estimations over generalized primes $p\in \mathcal{P}$.

Lemma 11.

Suppose that the estimate (5) is valid. Then, for $x\to \infty$,

$$\underset{p\in \mathcal{P}}{\sum _{p⩽x}}\frac{1}{p}=loglogx+a+O\left(x^{{\beta }_2-1}\right),$$

where a is a constant, and $0⩽{\beta }_2<1$.

Proof. 

We have

$$\begin{array}{cc}\hfill {\psi }_1\left(x\right)& \stackrel{\mathrm{def}}{=}\underset{p\in \mathcal{P}}{\sum _{p⩽x}}logp=\psi \left(x\right)-\underset{p\in \mathcal{P}}{\sum _{p^{\alpha }⩽x}}\sum _{2⩽\alpha ⩽(logx)/(log2)}logp\hfill \\ & =\psi \left(x\right)+O\left(\psi \left(x^{1/2}\right)logx\right)=x+r\left(x\right),\hfill \end{array}$$

where

$$r\left(x\right)=O\left(x^{{\beta }_2}logx\right)$$

with

$${\beta }_2=max\left({\beta }_1,\frac{1}{2}\right).$$

From this, by partial summation, we obtain

$$\begin{array}{cc}\hfill \underset{p\in \mathcal{P}}{\sum _{p⩽x}}\frac{1}{p}& =\frac{1}{xlogx}\underset{p\in \mathcal{P}}{\sum _{p⩽x}}logp+\underset{p_1}{\overset{x}{\int }}\left(\frac{1}{u^{2}logu}+\frac{1}{u^2{log}^{2}u}\right){\psi }_1\left(u\right)\mathrm{d}u\hfill \\ & =\frac{1}{logx}+loglogx-\frac{1}{logx}+c_1+\underset{p_1}{\overset{x}{\int }}\left(\frac{1}{u^{2}logu}+\frac{1}{u^2{log}^{2}u}\right)r\left(u\right)\mathrm{d}u\hfill \\ & =loglogx+c_1+\underset{p_1}{\overset{\infty }{\int }}\left(\frac{1}{u^{2}logu}+\frac{1}{u^2{log}^{2}u}\right)r\left(u\right)\mathrm{d}u\hfill \\ & -\underset{x}{\overset{\infty }{\int }}\left(\frac{1}{u^{2}logu}+\frac{1}{u^2{log}^{2}u}\right)r\left(u\right)\mathrm{d}u\hfill \\ & =loglogx+c_2+O\left(\underset{x}{\overset{\infty }{\int }}{u}^{{\beta }_1-2}\mathrm{d}u\right)=loglogx+c_2+O(\left(x^{{\beta }_2-1}\right).\hfill \end{array}$$

□

In what follows, we will use some properties of functions of exponential type. We recall a function $g\left(s\right)$ analytic in the region $|args|⩽{\theta }_0$, $0<{\theta }_0⩽\pi$ is of exponential type if uniformly in $\theta$, $\theta ⩽{\theta }_0$,

$$\underset{r\to \infty }{lim\; sup}\frac{log\left|g\right(r{\mathrm{e}}^{i\theta }\left)\right|}{r}<\infty .$$

Lemma 12.

Suppose that $g\left(s\right)$ is an entire function of exponential type, (5) holds, and

$$\underset{r\to \infty }{lim\; sup}\frac{log\left|g\right(r\left)\right|}{r}>-1.$$

Then

$$\sum _{p\in \mathcal{P}}\left|g(logp)\right|=\infty .$$

Proof. 

We use the formula of Lemma 11, and repeat word for word the proof of Theorem 6.4.14 of [22]. □

Let $s\in D$, and $|a_p|=1$. For brevity, we set

$$g_{\mathcal{P}}(s,a_p)=log\left(1-\frac{a_p}{p^s}\right),p\in \mathcal{P},$$

where

$$log\left(1-\frac{a_p}{p^s}\right)=-\frac{a_p}{p^s}-\frac{a_p^2}{2p^{2s}}-\cdots .$$

Lemma 13.

Suppose that (5) holds. Then the set of all convergent series

$$\sum _{p\in \mathcal{P}}{g}_{\mathcal{P}}(s,a_p)$$

is dense in the space $H\left(D\right)$.

Proof. 

The object connected to the system $\mathcal{P}$ is only Lemma 12. Other arguments of the proof are the same as those applied in the proof of Lemma 6.5.4 from [22]. □

Recall that the support of the distribution of a random element X is called a support of X, and is denoted by $S_X$.

For convenience, we state a lemma on the support of a series of random elements.

Lemma 14.

Let $\left\{{\xi }_m\right\}$ be a sequence of independent $H\left(D\right)$-valued random elements on a certain probability space $(\mathsf{\Xi },\mathcal{A},\mu )$; the series

$$\sum _{m=1}^{\infty }{\xi }_m$$

is convergent almost surely. Then, the support of the sum of this series is the closure of the set of all $g\in H\left(D\right)$ which may be written as a convergent series

$$g=\sum _{m=1}^{\infty }{g}_m,g_m\in S_{{\xi }_m}.$$

Proof. 

The lemma is Theorem 1.7.10 of [22], where its proof is given. □

Proof 

(Proof of Proposition 2). By the definition, $\left\{\omega \right(p):p\in \mathcal{P}\}$ is a sequence of independent complex-valued random variables. Therefore, $\left\{g_{\mathcal{P}}(s,\omega \left(p\right))\right\}$ is a sequence of independent $H\left(D\right)$-valued random elements. Since the support of each $\omega \left(p\right)$ is the unit circle, the support of $g_{\mathcal{P}}(s,\omega \left(p\right))\}$ is the set

$$\left\{g\in H\left(D\right):g\left(s\right)=-log\left(1-\frac{a}{p^s}\right),\left|a\right|=1\right\}.$$

Therefore, in view of Lemma 14, the support of the $H\left(D\right)$-valued random element

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

is the closure of the set of all convergent series

$$\sum _{p\in \mathcal{P}}{g}_{\mathcal{P}}(s,a_p)$$

with $|a_p|=1$. By Lemma 13, the set of the latter series is dense in $H\left(D\right)$. Define $u:H\left(D\right)\to H\left(D\right)$ by $u\left(g\right)={\mathrm{e}}^g$, $g\in H\left(D\right)$. The mapping u is continuous, $u(log{\zeta }_{\mathcal{P}}(s,\omega ))={\zeta }_{\mathcal{P}}(s,\omega )$ and $u\left(H\left(D\right)\right)=S_{\mathcal{P}}\setminus \left\{0\right\}$. This shows that $S_{\mathcal{P}}\setminus \left\{0\right\}$ lies in the support of ${\zeta }_{\mathcal{P}}(s,\omega )$. Since the support is a closed set, we obtain that the support of ${\zeta }_{\mathcal{P}}(s,\omega )$ contains the closure of $S_{\mathcal{P}}\setminus \left\{0\right\}$, i.e.,

$$S_{{\zeta }_{\mathcal{P}}}\supset S_{\mathcal{P}}.$$

(37)

On the other hand, the random element ${\zeta }_{\mathcal{P}}(s,\omega )$ is convergent for almost all $\omega \in {\mathsf{\Omega }}_{\mathcal{P}}$, a product of non-zeros multipliers. Therefore, by the classical Hurwitz theorem, see [29],

$$S_{{\zeta }_{\mathcal{P}}}\subset S_{\mathcal{P}}.$$

This inclusion together with (37) proves the proposition. □

7. Proof of Universality

In this section, we prove Theorem 2. Its proof is based on Theorem 3, Proposition 2 and the Mergelyan theorem [30] on the approximation of analytic functions by polynomials on compact sets with connected complements.

Proof of Theorem 2.

Let $p\left(s\right)$ be a polynomial, K and $\epsilon$ defined in Theorem 2, and

$${\mathcal{G}}_{\epsilon }=\left\{g\in H\left(D\right):\underset{s\in K}{sup}\left|g\left(s\right)-{\mathrm{e}}^{p\left(s\right)}\right|<\frac{\epsilon }{2}\right\}.$$

Then, the set ${\mathcal{G}}_{\epsilon }$ is an open neighborhood of an element ${\mathrm{e}}^{p\left(s\right)}\in S_{\mathcal{P}}$. Since, in view of Proposition 2, $S_{\mathcal{P}}$ is the support of the measure $P_{{\zeta }_{\mathcal{P}}}$, by a property of supports, we have

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

(38)

Since $f\left(s\right)\in H_0\left(K\right)$, we may apply the mentioned Mergelyan theorem and choose the polynomial $p\left(s\right)$ satisfying

$$\underset{s\in K}{sup}\left|f\left(s\right)-{\mathrm{e}}^{p\left(s\right)}\right|<\frac{\epsilon }{2}.$$

This shows that the set ${\mathcal{G}}_{\epsilon }$ lies in

$${\widehat{\mathcal{G}}}_{\epsilon }=\left\{g\in H\left(D\right):\underset{s\in K}{sup}|g\left(s\right)-f\left(s\right)|<\epsilon \right\}.$$

Thus, by (38), we have

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

(39)

Theorem 3 and the equivalent of weak convergence in terms of open sets yield

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

This, (39), and the definitions of $P_{T,\mathcal{P}}$ and ${\widehat{\mathcal{G}}}_{\epsilon }$ prove the first statement of the theorem.

To prove the second statement of the theorem, we observe that the boundary $\partial {\widehat{\mathcal{G}}}_{\epsilon }$ of the set ${\widehat{\mathcal{G}}}_{\epsilon }$ lies in the set

$$\left\{g\in H\left(D\right):\underset{s\in K}{sup}|g\left(s\right)-f\left(s\right)|=\epsilon \right\}.$$

Hence, the boundaries $\partial {\widehat{\mathcal{G}}}_{{\epsilon }_1}$ and $\partial {\widehat{\mathcal{G}}}_{{\epsilon }_2}$ do not intersect for different positive ${\epsilon }_1$ and ${\epsilon }_2$. Therefore, $P_{{\zeta }_{\mathcal{P}}}(\partial {\widehat{\mathcal{G}}}_{\epsilon })>0$ for countably many $\epsilon >0$. In other words, the set ${\widehat{\mathcal{G}}}_{\epsilon }$ is a continuity set of the measure $P_{{\zeta }_{\mathcal{P}}}$ for all but at most countably many $\epsilon >0$. This, (39), Theorem 3 and the equivalent of weak convergence in terms of continuity sets prove the second statement of the theorem. □

8. Conclusions

In the paper, we considered the set $\mathcal{P}$ of generalized prime numbers satisfying

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

and

$$\underset{m\in {\mathcal{N}}_{\mathcal{P}}}{\sum _{m⩽x}}{\mathsf{\Lambda }}_{\mathcal{P}}\left(m\right)=x+O\left(x^{{\beta }_1}\right),0⩽{\beta }_1<1,$$

where ${\mathcal{N}}_{\mathcal{P}}$ is the set of generalized integers and ${\mathsf{\Lambda }}_{\mathcal{P}}\left(m\right)$ is the generalized von Mangoldt function corresponding to the set $\mathcal{P}$. Assuming that the set $\{logp:p\in \mathcal{P}\}$ is linearly independent over $\mathbb{Q}$, and the Beurling zeta-function

$${\zeta }_{\mathcal{P}}\left(s\right)=\sum _{m\in {\mathcal{N}}_{\mathcal{P}}}\frac{1}{m^s},s=\sigma +it,\sigma >1,$$

has the bounded mean square for $\sigma >\widehat{\sigma }$ with some $\beta <\widehat{\sigma }<1$, we obtained universality of ${\zeta }_{\mathcal{P}}\left(s\right)$, i.e., that every non-vanishing analytic function can be approximated by shifts ${\zeta }_{\mathcal{P}}(s+i\tau )$, $\tau \in \mathbb{R}$.

In the future, we are planning to obtain a more complicated discrete version of Theorem 2, i.e., to prove the approximation of analytic functions by discrete shifts ${\zeta }_{\mathcal{P}}(s+ikh)$, $h>0$, $k\in {\mathbb{N}}_0$.