Let $\mathbb{P}$, $\mathbb{N}$, ${\mathbb{N}}_0$, $\mathbb{Z}$, $\mathbb{R}$, $\mathbb{C}$, as usual, denote the sets of primes, positive integers, nonnegative integers, integers, real and complex numbers, respectively, $s=\sigma +it$ a complex variable, $n\in \mathbb{N}$, Q a positive defined $n\times n$ matrix, and $Q\left[\underline{x}\right]={\underline{x}}^{\mathrm{T}}Q\underline{x}$ for $\underline{x}\in {\mathbb{Z}}^n$. In [1], Epstein considered a problem to find a zeta-function as general as possible and having the functional equation of the Riemann type. For $\sigma >\frac{n}{2}$, he defined the function

Now this function is called the Epstein zeta-function. It is analytically continuable to the whole complex plane, except for a simple pole at the point $s=\frac{n}{2}$ with residue where $\Gamma \left(s\right)$ is the Euler gamma-function. Epstein also proved that $\zeta (s;Q)$ satisfies the functional equation for all $s\in \mathbb{C}$.

It turned out that the Epstein zeta-function is an important number theoretical object having a series of practical applications, for example, in crystallography [2] and mathematical physics, more precisely, in quantum field theory and the Wheeler–DeWitt equation [3,4].

Value distribution of $\zeta (s;Q)$, as other zeta-functions, is not simple, and was studied by many authors including Hecke [5], Selberg [6], Iwaniec [7], Bateman [8], Fomenko [9], Pańkowski and Nakamura [10]. In [11] and [12], the characterization of asymptotic behaviour of $\zeta (s;Q)$ was given in terms of probabilistic limit theorems. The latter approach for the Riemann zeta-function was proposed by Bohr in [13], and realized in [14,15]. Denote by $\mathcal{B}\left(\mathbb{X}\right)$ the Borel $\sigma$-field of the space $\mathbb{X}$, and by measA the Lebesgue measure of a measurable set $A\subset \mathbb{R}$. For $A\in \mathcal{B}\left(\mathbb{C}\right)$, define

Under restrictions that $Q\left[\underline{x}\right]\in \mathbb{Z}$ for all $\underline{x}\in {\mathbb{Z}}^n\setminus \left\{0\right\}$, and $n\ge 4$ is even, it was obtained [11] that $P_{T,\sigma }^Q$, for $\sigma >\frac{n-1}{2}$, converges weakly to an explicitly given probability measure $P_{\sigma }^Q$ as $T\to \infty$. The discrete version of the latter theorem was given in [12].

The above restrictions on the matrix Q and [9], imply the decomposition with the zeta-function $\zeta (s;E_Q)$ of a certain Eisenstein series, and the zeta-function $\zeta (s;F_Q)$ of a certain cusp form.

Let $\chi$ be a Dirichlet character modulo q, and the corresponding Dirichlet L-function having analytic continuation to the whole complex plane if $\chi$ is nonprincipal character, and except for a simple pole at the point $s=1$ if $\chi$ is the principal character. Then (1), and [5,7], lead to the representation where ${\chi }_k$ and ${\widehat{\chi }}_l$ are Dirichlet characters, $a_{kl}\in \mathbb{C}$, $k,l\in \mathbb{N}$, and the series with coefficients $b_Q\left(m\right)$ converges absolutely in the half-plane $\sigma >\frac{n-1}{2}$. Thus, the investigation of the function $\zeta (s;Q)$ reduces to that of Dirichlet L-functions which, for $\sigma >1$, have the Euler product

Our aim is to describe by probabilistic terms the joint asymptotic behaviour of the function $\zeta (s;Q)$ and a zeta-function having no Euler’s product over primes. For this, the most suitable function is the classical Hurwitz zeta-function. Let $0<\alpha \le 1$ be a fixed parameter. The Hurwitz zeta-function $\zeta (s,\alpha )$ was introduced in [16], and is defined, for $\sigma >1$, by

Moreover, $\zeta (s,\alpha )$ has analytic continuation to the whole complex plane, except for a simple pole at the point $s=1$ with residue 1, $\zeta (s,1)=\zeta \left(s\right)$, and

Analytic properties of the function $\zeta (s,\alpha )$ depend on the arithmetic nature of the parameter $\alpha$. Some probabilistic limit theorems for the function $\zeta (s,\alpha )$ can be found, for example, in [17].

The statement of a joint limit theorem for the functions $\zeta (s;Q)$ and $\zeta (s,\alpha )$ requires some notations. Denote two tori

With the product topology and pointwise multiplication, ${\Omega }_1$ and ${\Omega }_2$ are compact topological Abelian groups. Therefore, again is a compact topological group. Hence, on $(\Omega ,\mathcal{B}(\Omega \left)\right)$, the probability Haar measure $m_H$ exists, and we have the probability space $(\Omega ,\mathcal{B}(\Omega ),m_H)$. Denote elements of $\Omega$ by $\omega =({\omega }_1,{\omega }_2)$, ${\omega }_1=({\omega }_1\left(p\right):p\in \mathbb{P})\in {\Omega }_1$ and ${\omega }_2=({\omega }_2\left(m\right):m\in {\mathbb{N}}_0)\in {\Omega }_2$, and, on the probability space $(\Omega ,\mathcal{B}(\Omega ),m_H)$ define, for ${\sigma }_1>\frac{n-1}{2}$ and ${\sigma }_2>\frac{1}{2}$, the ${\mathbb{C}}^2$-valued random element where $\underline{\sigma }=({\sigma }_1,{\sigma }_2)$, with and

Let

Moreover, denote by $P_{\underline{\zeta },\underline{\sigma }}$ the distribution of the random element $\underline{\zeta }(\underline{\sigma },\omega ,\alpha ;Q)$, i.e.,

The main the result of the paper is the following joint limit theorem of Bohr-Jessen type for the functions $\zeta (s;Q)$ and $\zeta (s,\alpha )$.

For brevity, we set

For example, if the parameter $\alpha$ is transcendental, then the set $L(\mathbb{P},\alpha )$ is linearly independent over $\mathbb{Q}$.

We divide the proof of Theorem 1 into several lemmas which are limit theorems in some spaces for certain auxiliary objects. The important place of the proof is the identification of the limit measure.

For $A\in \mathcal{B}(\Omega )$, set

Lemma 1 is a starting point for the proof of limit lemmas in ${\mathbb{C}}^2$ for certain objects given by absolutely convergent Dirichlet series.

Fix $\beta >\frac{1}{2}$, and, for $N\in \mathbb{N}$, set and

Define and

Since $u_N\left(m\right)$ and $u_N(m,\alpha )$ decrease with respect to m exponentially, the above series are absolutely convergent for $\sigma >{\sigma }_0$ with arbitrary fixed finite ${\sigma }_0$. For ${\sigma }_1>\frac{n-1}{2}$ and ${\sigma }_2>\frac{1}{2}$, let with and with

For $A\in \mathcal{B}\left({\mathbb{C}}^2\right)$, define and

This section is devoted to weak convergence of $P_{T,N,\underline{\sigma }}^{Q,\alpha }$ and $P_{T,N,\underline{\sigma }}^{Q,\alpha ,\Omega }$ as $T\to \infty$. Let the mapping $v_{N,\underline{\sigma }}^{Q,\alpha }:\Omega \to {\mathbb{C}}^2$ be given by and $V_{N,\underline{\sigma }}^{Q,\alpha }=m_H{\left(v_{N,\underline{\sigma }}^{Q,\alpha }\right)}^{-1}$, where, for $A\in \mathcal{B}\left({\mathbb{C}}^2\right)$,

Since all Dirichlet series in the definition of ${\underline{\zeta }}_N(\underline{\sigma },\omega ,\alpha ;Q)$ are absolutely convergent in the considered region, the mapping $v_{N,\underline{\sigma }}^{Q,\alpha }$ is continuous, hence $(\mathcal{B}(\Omega ),\mathcal{B}\left({\mathbb{C}}^2\right))$-measurable. Therefore, the probability measure $V_{N,\underline{\sigma }}^{Q,\alpha }$ is defined correctly, see, for example, [18], Section 5.

In this section, we approximate $\underline{\zeta }(\underline{\sigma }+it,\alpha ;Q)$ by ${\underline{\zeta }}_N(\underline{\sigma }+it,\alpha ;Q)$ and $\underline{\zeta }(\underline{\sigma }+it,\omega ,\alpha ;Q)$ by ${\underline{\zeta }}_N(\underline{\sigma }+it,\omega ,\alpha ;Q)$.

Let, for ${\underline{z}}_1=(z_{11},z_{12})$, ${\underline{z}}_2=(z_{21},z_{22})\in {\mathbb{C}}^2$,

Let $\left\{P\right\}$ be a family of probability measures on $(\mathbb{X},\mathcal{B}(\mathbb{X}\left)\right)$. Remind that the family $\left\{P\right\}$ is called tight if, for every $ϵ>0$, there exists a compact set $K\subset \mathbb{X}$ such that for all $P\in \left\{P\right\}$. The family $\left\{P\right\}$ is relatively compact if every sequence $\left\{P_n\right\}\subset \left\{P\right\}$ contains a subsequence $\left\{P_n\right\}$ weakly convergent to a certain probability measure on $(\mathbb{X},\mathcal{B}(\mathbb{X}\left)\right)$ as $n\to \infty$.

A property of relative compactness is useful for investigation of weak convergence of probability measures. By the classical Prokhorov theorem, see, for example, [18], every tight family $\left\{P\right\}$ is relatively compact as well. Therefore, often it is convenient to know the tightness of considered family. In our case, this concerns the measure $V_N^{Q,\alpha }$, $N\in \mathbb{N}$.

Now we are ready to prove weak convergence for $P_{T,\underline{\zeta },\underline{\sigma }}$ and