A New Joint Limit Theorem of Bohr–Jessen Type for Epstein Zeta-Functions

Abstract

For $j=1,\dots ,r$, let $Q_j$ be a positive definite $n_j\times n_j$ matrix, and $\zeta (s_j;Q_j)$ denote the corresponding Epstein zeta-function. In this paper, assuming that $n_j⩾4$ is even and ${\underline{x}}^{\mathrm{T}}{Q}_j\underline{x}\in \mathbb{Z}$,$\underline{x}\in {\mathbb{Z}}^r\setminus \left\{\underline{0}\right\}$, a joint limit theorem of Bohr–Jessen type for the functions $\zeta (s_1;Q_1),\dots ,\zeta (s_r;Q_r)$, by using generalizing shifts $\zeta ({\sigma }_1+i{\phi }_1\left(t\right);Q_1),\dots ,\zeta ({\sigma }_r+i{\phi }_r\left(t\right);Q_r)$, is proved. Here, the functions ${\phi }_1\left(t\right),\dots ,{\phi }_r\left(t\right)$ are increasing to $+\infty$, with monotonic derivatives ${\phi }_j^{\prime }\left(t\right)$ satisfying the asymptotic growth conditions: ${\phi }_j\left(t\right)\ll t{\phi }_j^{\prime }\left(t\right)$, and ${\phi }_j^{\prime }\left(t\right)=o\left({\phi }_{j+1}^{\prime }\left(t\right)\right)$ as $t\to \infty$. An explicit form of the limit measure is given. This theorem extends and generalizes the previous result on the joint value-distribution of Epstein zeta-functions.

1. Introduction

Let $\mathbb{P},2\mathbb{N},\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$ denote the sets of all prime, positive even integer, positive integer, integer, rational, real and complex numbers, respectively, and $s=\sigma +it\in \mathbb{C}$. Moreover, let Q be a positive definite $n\times n$, $n\in \mathbb{N}$, matrix and $Q\left[\underline{x}\right]={\underline{x}}^{\mathrm{T}}Q\underline{x}$, $\underline{x}\in {\mathbb{Z}}^n=\underset{n}{\underbrace{\mathbb{Z}\times \dots \times \mathbb{Z}}}$. The Epstein zeta-function $\zeta (s;Q)$ is defined, for $\sigma >\frac{n}{2}$, by the series

$$\zeta (s;Q)=\sum _{\underline{x}\in {\mathbb{Z}}^n\setminus \left\{\underline{0}\right\}}{\left(Q\left[\underline{x}\right]\right)}^{-s},$$

and has the analytic continuation to the whole complex plane, except for the point $s=\frac{n}{2}$ which is a simple pole with residue ${\pi }^{n/2}{\left(\mathsf{\Gamma }\left(\frac{n}{2}\right)\sqrt{\det Q}\right)}^{-1}$, where $\mathsf{\Gamma }\left(s\right)$ is the Euler gamma-function. The function $\zeta (s;Q)$ was introduced by P. Epstein [1] with the aim of generalizing the Riemann zeta-function

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

and its functional equation

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

(1)

Clearly, for $n=1$ and $Q=\left(1\right)$, we have $\zeta (s;Q)=2\zeta \left(2s\right)$. Epstein’s attempt was successful, and he obtained the functional equation for $\zeta (s;Q)$:

$${\pi }^{-s}\mathsf{\Gamma }\left(s\right)\zeta (s;Q)=\sqrt{\det Q}{\pi }^{s-{\displaystyle \frac{n}{2}}}\mathsf{\Gamma }\left({\displaystyle \frac{n}{2}}-s\right)\zeta \left({\displaystyle \frac{n}{2}}-s;Q^{-1}\right),$$

which, as in (1), is valid for all $s\in \mathbb{C}$, and $Q^{-1}$ denotes the inverse matrix of Q. This and (1) show that $\zeta \left(s\right)$ has the symmetric functional equation, while in the functional equation for $\zeta (s;Q)$, a new function $\zeta (s;Q^{-1})$ appears, but symmetry with respect to s is preserved. Although the functions $\zeta \left(s\right)$ and $\zeta (s;Q)$ have functional equations of the same Riemann type, their properties are quite different. For example, the function $\zeta \left(s\right)\ne 0$ in the half-plane of absolute convergence $\sigma >1$, while there exist matrices Q such that $\zeta (s;Q)$ has infinitely many zeros in the half plane $\sigma >\frac{n}{2}$. Zero distribution of $\zeta (s;Q)$ is also a significant problem, comparable to that of $\zeta \left(s\right)$, and has been studied by numerous authors. We mention some results here. It is known that, for certain matrices, the Riemann hypothesis for $\zeta (s;Q)$ does not hold; there exist zeros of $\zeta (s;Q)$ off the critical line $\sigma =\frac{n}{4}$ [2]. Moreover, it was shown in [3] that, differently from the case of $\zeta \left(s\right)$, the zeros of $\zeta (s;Q)$ are generally not symmetric with respect to the line $\sigma =\frac{n}{4}$. Estimates for the number of zeros in the strips have been studied by E. Bobmbieri and J. Mueller [4], Y Lee [5], and others. Also, it is known [6] that imaginary parts of the zeros of Epstein zeta-functions are uniformly distributed modulo 1. Recently, an interesting formula for the sum of values of $\zeta (s;Q)$ over the nontrivial zeros of $\zeta \left(s\right)$ was proved in [7]. Thus, Epstein provided mathematicians with a novel object of algebraic and analytic nature, which has stimulated extensive research in number theory and related fields.

The function $\zeta (s;Q)$ is an automorphic form with respect to an unimodular group; it appears in the problems of algebraic number theory. It also has a range of practical applications, including crystallography [8], quantum field theory [9,10] and temperature and energy problems [11,12,13,14]. In general, the Epstein zeta-function is an attractive analytical object and is widely studied.

Unfortunately, we do not know any monograph devoted to classical results on the function $\zeta (s;Q)$. Some desired results can be found in the works on automorphic forms; see, for example, [15,16].

In [17], we began to characterize the asymptotic behaviour of the function $\zeta (s;Q)$ by using the Bohr–Jessen method [18,19], and techniques developed in [20]. Note that H. Bohr and B. Jessen considered only the existence of density on certain sets (rectangles) for the Riemann zeta-function, without giving an explicit form. Denote by $\mathcal{B}\left(\mathbb{X}\right)$ the Borel $\sigma$-field of the space $\mathbb{X}$, and by meas A the Lebesgue measure of a measurable set $A\subset \mathbb{R}$. Then, the asymptotic behaviour of $\zeta (s;Q)$ can be described by the asymptotics of

$${\displaystyle \frac{1}{T}}\mathrm{meas}\left\{t\in [0,T]:\zeta (\sigma +it;Q)\in A\right\},A\in \mathcal{B}\left(\mathbb{C}\right),$$

as $T\to \infty$. For this, it is convenient to use the weak convergence of probability measures.

Really, the function $\zeta (s;Q)$ is a class of Dirichlet series depending on the matrix Q. This class is rather general in obtaining results that are full of sense. In order for the function $\zeta (s;Q)$ to be close to number-theoretical objects, it is sufficient to limit ourself by matrices Q for which $Q\left[\underline{x}\right]\in \mathbb{Z}$ for all $\underline{x}\in {\mathbb{Z}}^n\setminus \left\{\underline{0}\right\}$. In this case, the function $\zeta (s;Q)$, for $\sigma >\frac{n}{2}$, can be expressed in the following form [21]:

$$\zeta (s;Q)=\zeta (s;E_Q)+\zeta (s;F_Q),$$

where $\zeta (s;E_Q)$ and $\zeta (s;F_Q)$ are corresponding zeta-functions of a certain Eisenstein series and modular forms of weight $\frac{n}{2}$, respectively. Moreover, it is convenient to additionally require that $n\in 2\mathbb{N}$ and $n⩾4$. Then, $\zeta (s;Q)$ is a combination of products of Dirichlet L-functions and an absolutely convergent Dirichlet series [15,16]. More precisely, let $q\in \mathbb{N}$ be such that $q{\left(2Q\right)}^{-1}$ is an integral matrix, k and l positive divisors of q, ${\chi }_k$ and ${\chi }_l$ Dirichlet characters modulo $q/k$ and $q/l$, respectively, and

$$L(s,{\chi }_k)=\sum _{m=1}^{\infty }{\displaystyle \frac{{\chi }_k\left(m\right)}{m^s}},L(s,{\psi }_l)=\sum _{m=1}^{\infty }{\displaystyle \frac{{\psi }_l\left(m\right)}{m^s}},\sigma >1,$$

the corresponding Dirichlet L-functions. Then, for $\sigma >\frac{n-1}{2}$,

$$\zeta (s;Q)=\sum _{k=1}^K\sum _{l=1}^L{\displaystyle \frac{a_{kl}}{k^s{l}^s}}L(s,{\chi }_k)L\left(s-{\displaystyle \frac{n}{2}}+1,{\psi }_l\right)+\sum _{m=1}^{\infty }{\displaystyle \frac{f_Q\left(m\right)}{m^s}},$$

(2)

where $a_{kl}\in \mathbb{C}$ are certain numbers, the characters ${\chi }_k$, $1\le k\le K$, are pairwise nonequivalent, and ${\chi }_l$, $1\le l\le L$, are pairwise nonequivalent too, and the Dirichlet series is absolutely convergent in the half-plane $\sigma >\frac{n-1}{2}$. In view of equality (2), the investigations of the function $\zeta (s;Q)$, under the above hypotheses, reduce to those of Dirichlet L-functions.

For the definition of the limit measure in a limit theorem for $\zeta (s;Q)$, the set

$$\mathsf{\Omega }=\prod _{p\in \mathbb{P}}{\gamma }_p,$$

where ${\gamma }_p=\{s\in \mathbb{C}:|s|=1\}$, for all $p\in \mathbb{P}$, plays a crucial role. The set $\mathsf{\Omega }$ consists of all functions from $\mathbb{P}$ into the unit circle. With the product topology and pointwise multiplication, the torus $\mathsf{\Omega }$ is a compact topological group; therefore, on $(\mathsf{\Omega },\mathcal{B}(\mathsf{\Omega }\left)\right)$, the probability Haar measure $m_{1H}$ exists, and we arrive at the probability space $(\mathsf{\Omega },\mathcal{B}\left(\mathsf{\Omega }\right),m_{1H})$. Let $\omega \left(p\right)$ be the projection of $\omega \in \mathsf{\Omega }$ to the coordinate space ${\gamma }_p$, $p\in \mathbb{P}$. Extend the function $\omega \left(p\right)$, $p\in \mathbb{P}$, to the set $\mathbb{N}$ by using the formula

$$\omega \left(m\right)=\prod _{p^{\alpha }\Vert m}{\omega }^{\alpha }\left(p\right),m\in \mathbb{N},$$

where $p^{\alpha }\Vert m$ means that $p^{\alpha }\mid m$, but $p^{\alpha +1}\nmid m$. For an arbitrary Dirichlet L-function $L(s,\chi )$, define

$$L(s,\omega ,\chi )=\sum _{m=1}^{\infty }{\displaystyle \frac{\chi \left(m\right)\omega \left(m\right)}{m^s}}.$$

The latter series, for almost all $\omega \in \mathsf{\Omega }$, converges in the half-pane $\sigma >\frac{1}{2}$, and is a complex-valued random element on $(\mathsf{\Omega },\mathcal{B}\left(\mathsf{\Omega }\right),m_{1H})$. Moreover, for almost all $\omega \in \mathsf{\Omega }$, the equality

$$L(s,\omega ,\chi )=\prod _{p\in \mathbb{P}}{\left(1-{\displaystyle \frac{\chi \left(p\right)\omega \left(p\right)}{p^s}}\right)}^{-1},\sigma >{\displaystyle \frac{1}{2}},$$

is valid. For $\sigma >{\displaystyle \frac{n-1}{2}}$, set

$$\begin{array}{ccc}\hfill \zeta (\sigma ,\omega ;Q)& =& \sum _{k=1}^K\sum _{l=1}^L{\displaystyle \frac{a_{kl}\omega \left(k\right)\omega \left(l\right)}{k^{\sigma }{l}^{\sigma }}}L(\sigma ,\omega ,{\chi }_k)L\left(\sigma -{\displaystyle \frac{n}{2}}+1,\omega ,{\psi }_l\right)\hfill \\ & +& \sum _{m=1}^{\infty }{\displaystyle \frac{f_Q\left(m\right)\omega \left(m\right)}{m^{\sigma }}}.\hfill \end{array}$$

Then, $\zeta (\sigma ,\omega ;Q)$ is a complex-valued random element on $(\mathsf{\Omega },\mathcal{B}\left(\mathsf{\Omega }\right),m_{1H})$, and let

$$P_{\sigma ;Q}\left(A\right)=m_{1H}\left\{\omega \in \mathsf{\Omega }:\zeta (\sigma ,\omega ;Q)\in A\right\},A\in \mathcal{B}\left(\mathbb{C}\right),$$

be its distribution. The main result of [17] is the following theorem on the weak convergence for

$$P_{T,\sigma ;Q}\left(A\right)={\displaystyle \frac{1}{T}}\mathrm{meas}\left\{t\in [0,T]:\zeta (\sigma +it;Q)\in A\right\},A\in \mathcal{B}\left(\mathbb{C}\right).$$

Theorem 1.

Suppose that $\sigma >{\displaystyle \frac{n-1}{2}}$ is fixed. Then, $P_{T,\sigma ;Q}$ converges weakly to the measure $P_{\sigma ;Q}$ as $T\to \infty$.

In [22], a joint version of Theorem 1 has been obtained. For $j=1,\dots ,r$, let $Q_j$ be a positive definite quadratic $n_j\times n_j$ matrix, and $\zeta (s_j;Q_j)$ be the corresponding Epstein zeta-function. Denote $\underline{s}=(s_1,\dots ,s_r)$, $\underline{Q}=(Q_1,\dots ,Q_r)$ and $\underline{\zeta }(\underline{s};\underline{Q})=(\zeta (s_1;Q_1),\dots ,\zeta (s_r;Q_r))$. On the probability space $(\mathsf{\Omega },\mathcal{B}\left(\mathsf{\Omega }\right),m_{1H})$, define the ${\mathbb{C}}^r$-valued (${\mathbb{C}}^r=\underset{r}{\underbrace{\mathbb{C}\times \dots \times \mathbb{C}}}$) random element

$$\underline{\zeta }(\underline{\sigma },\omega ;\underline{Q})=\left(\zeta ({\sigma }_1,\omega ;Q_1),\dots ,\zeta ({\sigma }_r,\omega ;Q_r)\right),$$

where ${\sigma }_j>{\displaystyle \frac{n_j-1}{2}}$, and

$$\begin{array}{ccc}\hfill \zeta ({\sigma }_j,\omega ;Q_j)& =& \sum _{k=1}^{K_j}\sum _{l=1}^{L_j}{\displaystyle \frac{a_{klj}\omega \left(k\right)\omega \left(l\right)}{k^{{\sigma }_j}{l}^{{\sigma }_j}}}L({\sigma }_j,\omega ,{\chi }_{kj})L\left({\sigma }_j-{\displaystyle \frac{n_j}{2}}+1,\omega ,{\psi }_{lj}\right)\hfill \\ & & +\sum _{m=1}^{\infty }{\displaystyle \frac{f_{Q_j}\left(m\right)\omega \left(m\right)}{m^{{\sigma }_j}}},\hfill \end{array}$$

with corresponding $a_{klj}\in \mathbb{C}$, $K_j\in \mathbb{N}$, $L_j\in \mathbb{N}$, and Dirichlet characters ${\chi }_{kj}$ and ${\psi }_{lj}$, $j=1,\dots ,r$.

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

$${\widehat{P}}_{\underline{\zeta }}\left(A\right)={\widehat{P}}_{\underline{\zeta },\underline{\sigma };\underline{Q}}\left(A\right)=m_{1H}\left\{\omega \in \mathsf{\Omega }:\underline{\zeta }(\underline{\sigma },\omega ;\underline{Q})\in A\right\}$$

and

$${\widehat{P}}_T\left(A\right)={\widehat{P}}_{T,\underline{\sigma };\underline{Q}}\left(A\right)={\displaystyle \frac{1}{T}}\mathrm{meas}\left\{t\in [0,T]:\underline{\zeta }(\underline{\sigma }+it;\underline{Q})\in A\right\}.$$

Then, in [22], the following limit theorem has been given.

Theorem 2.

Suppose that ${\sigma }_j>{\displaystyle \frac{n_j-1}{2}}$ is fixed, where $j=1,\dots ,r$. Then, ${\widehat{P}}_T$ converges weakly to the measure ${\widehat{P}}_{\underline{\zeta }}$ as $T\to \infty$.

In [23], a generalization of Theorem 1 has been given, i. e., the weak convergence for

$${\displaystyle \frac{1}{T}}\mathrm{meas}\left\{t\in [T,2T]:\zeta (\sigma +i\phi (t);Q)\in A\right\},A\in \mathcal{B}\left(\mathbb{C}\right),$$

with a certain differentiable function $\phi \left(t\right)$ has been obtained as $T\to \infty$. The aim of this paper is to prove a joint version of the above-mentioned theorem from [23]. We note that using generalized shifts $\zeta (\sigma +i\phi (t);Q)$ allows for the more complete characterization of the asymptotic behaviour of the function $\zeta (s;Q)$.

Let $T_0>0$ be a fixed sufficiently large number. We say that a collection of real-valued functions $({\phi }_1\left(t\right),\dots ,{\phi }_r\left(t\right))$ defined for $t⩾T_0$ belongs to the class $U_r\left(T_0\right)$ if the following conditions are satisfied:

$1^{\circ }$ for every $j=1,\dots ,r$, ${\phi }_j\left(t\right)$ is an increasing to $+\infty$ function;

$2^{\circ }$ for every $j=1,\dots ,r$, ${\phi }_j\left(t\right)$ has a monotonic derivative ${\phi }_j^{\prime }\left(t\right)$ such that

$${\phi }_j\left(t\right)\ll t{\phi }_j^{\prime }\left(t\right);$$

$3^{\circ }$ for every $j=2,\dots ,r$ and $k\le j$, ${\phi }_k^{\prime }\left(t\right)=o\left({\phi }_j^{\prime }\left(t\right)\right)$ as $t\to \infty$.

For example, we can take ${\phi }_j\left(t\right)=t^{\alpha +j}$ with fixed $\alpha >0$. We recall that $a{\ll }_{\theta }b$, $b>0$, means that there exists a constant $c=c\left(\theta \right)>0$ such that $\left|a\right|\le cb$.

For the statement of a joint limit theorem with shifts $\zeta (\sigma +i{\phi }_j\left(t\right);Q)$, we need a new probability space. Let $\mathsf{\Omega }$ be the same group as above. Define

$${\mathsf{\Omega }}^r={\mathsf{\Omega }}_1\times \dots \times {\mathsf{\Omega }}_r,$$

where ${\mathsf{\Omega }}_j=\mathsf{\Omega }$ for all $j=1,\dots ,r$. Then, by the classical Tikhonov theorem, ${\mathsf{\Omega }}^r$ is a compact topological group. Therefore, on $({\mathsf{\Omega }}^r,\mathcal{B}\left({\mathsf{\Omega }}^r\right))$, the probability Haar measure $m_H$ can be defined. Note that the measure $m_H$ is the product of the Haar measures $m_{jH}$ on $({\mathsf{\Omega }}_j,\mathcal{B}\left({\mathsf{\Omega }}_j\right))$, $j=1,\dots ,r$. Thus, we have the probability space $({\mathsf{\Omega }}^r,\mathcal{B}\left({\mathsf{\Omega }}^r\right),m_H)$. Denote by ${\omega }_j$ the elements of ${\mathsf{\Omega }}_j$ and by $\omega =\left({\omega }_1,\dots ,{\omega }_r\right)$ the elements of ${\mathsf{\Omega }}^r$. Now, on the probability space $({\mathsf{\Omega }}^r,\mathcal{B}\left({\mathsf{\Omega }}^r\right),m_H)$, the ${\mathbb{C}}^r$-valued random element

$$\underline{\zeta }(\underline{\sigma },\omega ;\underline{Q})=\left(\zeta ({\sigma }_1,{\omega }_1;Q_1),\dots ,\zeta ({\sigma }_r,{\omega }_r;Q_r)\right),$$

is defined, where ${\sigma }_j>{\displaystyle \frac{n_j-1}{2}}$, and

$$\begin{array}{ccc}\hfill \zeta ({\sigma }_j,{\omega }_j;Q_j)& =& \sum _{k=1}^{K_j}\sum _{l=1}^{L_j}{\displaystyle \frac{a_{klj}{\omega }_j\left(k\right){\omega }_j\left(l\right)}{k^{{\sigma }_j}{l}^{{\sigma }_j}}}L({\sigma }_j,{\omega }_j,{\chi }_{kj})L\left({\sigma }_j-{\displaystyle \frac{n_j}{2}}+1,{\omega }_j,{\psi }_{lj}\right)\hfill \\ & +& \sum _{m=1}^{\infty }{\displaystyle \frac{f_{Q_j}\left(m\right){\omega }_j\left(m\right)}{m^{{\sigma }_j}}},j=1,\dots ,r.\hfill \end{array}$$

Let $P_{\underline{\zeta }}$ be the distribution of the random element $\underline{\zeta }\left(\underline{\sigma },\omega ;\underline{Q}\right)$, i.e.,

$$P_{\underline{\zeta }}\left(A\right)=P_{\underline{\zeta };\underline{\sigma },\underline{Q}}\left(A\right)=m_H\left\{\omega \in {\mathsf{\Omega }}^r:\underline{\zeta }(\underline{\sigma },\omega ;\underline{Q})\in A\right\},A\in \mathcal{B}\left({\mathbb{C}}^r\right).$$

Define

$$P_T\left(A\right)=P_{T,\underline{\sigma };\underline{Q}}\left(A\right)={\displaystyle \frac{1}{T}}\mathrm{meas}\left\{t\in [T,2T]:\underline{\zeta }(\underline{\sigma }+i\underline{\phi }\left(t\right);\underline{Q})\in A\right\},A\in \mathcal{B}\left({\mathbb{C}}^r\right),$$

where $\underline{\phi }\left(t\right)=\left({\phi }_1\left(t\right),\dots ,{\phi }_r\left(t\right)\right)$, and

$$\underline{\zeta }(\underline{\sigma }+i\underline{\phi }\left(t\right);\underline{Q})=\left(\zeta ({\sigma }_1+i{\phi }_1\left(t\right);Q_1),\dots ,\zeta ({\sigma }_r+i{\phi }_r\left(t\right);Q_r)\right)$$

with

$$\begin{array}{ccc}\hfill \zeta ({\sigma }_j+i{\phi }_j\left(t\right);Q_j)& =& \sum _{k=1}^{K_j}\sum _{l=1}^{L_j}{\displaystyle \frac{a_{klj}}{k^{{\sigma }_j+i{\phi }_j\left(t\right)}{l}^{{\sigma }_j+i{\phi }_j\left(t\right)}}}L({\sigma }_j+i{\phi }_j\left(t\right),{\chi }_{kj})\hfill \\ & & ·L\left({\sigma }_j+i{\phi }_j\left(t\right)-{\displaystyle \frac{n_j}{2}}+1,{\psi }_{lj}\right)\hfill \\ & +& \sum _{m=1}^{\infty }{\displaystyle \frac{f_{Q_j}\left(m\right)}{m^{{\sigma }_j+i{\phi }_j\left(t\right)}}}.\hfill \end{array}$$

Theorem 3.

Suppose that $\left({\phi }_1\left(t\right),\dots ,{\phi }_r\left(t\right)\right)\in U_r\left(T_0\right)$, and ${\sigma }_j>{\displaystyle \frac{n_j-1}{2}}$ is fixed, where $j=1,\dots ,r$. Then, $P_T$ converges weakly to the measure $P_{\underline{\zeta }}$ as $T\to \infty$.

Thus, Theorem 3 provides a joint extension of Theorem 2. We emphasize the importance of condition $3^{\circ }$ in the definition of the class ${\mathcal{U}}_r\left(T_0\right)$. It is important to mention that probabilistic limit theorems accurately reflect the chaotic behaviour of the functions $\zeta (s_1;Q_1),\dots ,\zeta (s_r;Q_r)$, and can be applied to further investigations related to approximation problems.

The proof of Theorem 3 is divided into parts. First, the weak convergence on ${\mathsf{\Omega }}^r$ is established. Next, some absolutely convergent Dirichlet series are considered, and finally, the assertion of Theorem 3 is proved.

2. Case of ${\mathsf{\Omega }}^{\mathbf{r}}$

For $A\in \mathcal{B}\left({\mathsf{\Omega }}^r\right)$, define

$$\begin{array}{ccc}\hfill R_T\left(A\right)& =& R_{T,{\mathsf{\Omega }}^r,\underline{\phi }}\left(A\right)\hfill \\ & =& {\displaystyle \frac{1}{T}}\mathrm{meas}\left\{t\in [T,2T]:\left(\left(p^{-i{\phi }_1\left(t\right)}:p\in \mathbb{P}\right),\dots ,\left(p^{-i{\phi }_r\left(t\right)}:p\in \mathbb{P}\right)\right)\in A\right\}.\hfill \end{array}$$

Lemma 1.

Suppose that $\left({\phi }_1\left(t\right),\dots ,{\phi }_r\left(t\right)\right)\in U_r\left(T_0\right)$. Then, $R_T$ converges weakly to the Haar measure $m_H$ as $T\to \infty$.

Proof. 

We have to prove that the Fourier transform $f_T({\underline{k}}_1,\dots ,{\underline{k}}_r)=f_{T,{\mathsf{\Omega }}^r,\underline{\phi }}({\underline{k}}_1,\dots ,{\underline{k}}_r)$ of $R_T$ (where ${\underline{k}}_j=\left(k_{jp}:k_{jp}\in \mathbb{Z},p\in \mathbb{P}\right)$, $j=1,...,r$) converges to the Fourier transform

$$f({\underline{k}}_1,\dots ,{\underline{k}}_r)=\left\{\begin{array}{c}1\mathrm{if}({\underline{k}}_1,\dots ,{\underline{k}}_r)=(\underline{0},\dots ,\underline{0}),\hfill \\ 0\mathrm{if}({\underline{k}}_1,\dots ,{\underline{k}}_r)\ne (\underline{0},\dots ,\underline{0}),\hfill \end{array}\right.$$

of the Haar measure $m_H$ as $T\to \infty$. Here, $\underline{0}$ denotes the collection of zeros.

By the definition of $R_T$, we have

$$\begin{array}{ccc}\hfill f_T({\underline{k}}_1,\dots ,{\underline{k}}_r)& =& \underset{{\mathsf{\Omega }}^r}{\int }\left(\prod _{j=1}^r{\prod _{p\in \mathbb{P}}}^{*}{\omega }_j^{k_{jp}}\left(p\right)\right)\mathrm{d}{R}_T={\displaystyle \frac{1}{T}}\underset{T}{\overset{2T}{\int }}\left(\prod _{j=1}^r{\prod _{p\in \mathbb{P}}}^{*}{p}^{-i{k}_{jp}{\phi }_j\left(t\right)}\right)\mathrm{d}t\hfill \\ & =& {\displaystyle \frac{1}{T}}\underset{T}{\overset{2T}{\int }}exp\left\{-i\sum _{j=1}^r{\phi }_j\left(t\right){\sum _{p\in \mathbb{P}}}^{*}{k}_{jp}logp\right\}\mathrm{d}t,\hfill \end{array}$$

(3)

where the star “*” shows that only a finite number of integers $k_{jp}$ are distinct from zero. Obviously,

$$f_T(\underline{0},\dots ,\underline{0})=1.$$

(4)

Thus, it remains to consider the case $({\underline{k}}_1,\dots ,{\underline{k}}_r)\ne (\underline{0},\dots ,\underline{0})$. Set

$$\begin{array}{c}\hfill A\left(t\right)=\sum _{j=1}^r{\phi }_j\left(t\right){\sum _{p\in \mathbb{P}}}^{*}{k}_{jp}logp=\sum _{j=1}^r{\kappa }_j{\phi }_j\left(t\right),\end{array}$$

where

$${\kappa }_j={\sum _{p\in \mathbb{P}}}^{*}{k}_{jp}logp.$$

It is well known that the set of logarithms of prime numbers is linearly independent over $\mathbb{Q}$. Therefore, there exist $j\in \{1,\dots ,r\}$ such that ${\kappa }_j\ne 0$. Let $j_0=max(j:{\kappa }_j\ne 0)$. Then, by the definition of the class $U_r\left(T_0\right)$, we have, for $j⩽j_0$,

$$\begin{array}{c}\hfill A^{\prime }\left(t\right)=\sum _{j⩽j_0}{\kappa }_j{\phi }_j^{\prime }\left(t\right)={\kappa }_{j_0}{\phi }_{j_0}^{\prime }\left(t\right)(1+o\left(1\right)),t\to \infty .\end{array}$$

Hence, by the identity

$${\displaystyle \frac{1}{1+a}}=1-{\displaystyle \frac{a}{1+a}},a\ne -1,$$

we find

$${\displaystyle \frac{1}{A^{\prime }\left(t\right)}}={\displaystyle \frac{1}{{\kappa }_{j_0}{\phi }_{j_0}^{\prime }\left(t\right)}}(1+o\left(1\right)),t\to \infty .$$

Therefore,

$$\begin{array}{ccc}\hfill \underset{T}{\overset{2T}{\int }}cos\left(A\left(t\right)\right)\mathrm{d}t& =& \underset{T}{\overset{2T}{\int }}{\displaystyle \frac{1}{A^{\prime }\left(t\right)}}cos\left(A\left(t\right)\right)\mathrm{d}\left(A\left(t\right)\right)\hfill \\ & =& {\displaystyle \frac{1}{{\kappa }_{j_0}}}\underset{T}{\overset{2T}{\int }}{\displaystyle \frac{1}{{\phi }_{j_0}^{\prime }\left(t\right)}}cos\left(A\left(t\right)\right)\mathrm{d}\left(A\left(t\right)\right)+{\displaystyle \frac{1}{{\kappa }_{j_0}}}\underset{T}{\overset{2T}{\int }}{\displaystyle \frac{o\left(1\right)}{{\phi }_{j_0}^{\prime }\left(t\right)}}cos\left(A\left(t\right)\right)\mathrm{d}\left(A\left(t\right)\right)\hfill \\ & =& {\displaystyle \frac{1}{{\kappa }_{j_0}}}\underset{T}{\overset{2T}{\int }}{\displaystyle \frac{1}{{\phi }_{j_0}^{\prime }\left(t\right)}}\mathrm{d}(sin\left(A\left(t\right)\right))+\underset{T}{\overset{2T}{\int }}{\displaystyle \frac{o\left(1\right)(1+o(1\left)\right)}{A^{\prime }\left(t\right)}}\mathrm{d}(sin\left(A\left(t\right)\right))\hfill \\ & =& {\displaystyle \frac{1}{{\kappa }_{j_0}}}\underset{T}{\overset{2T}{\int }}{\displaystyle \frac{1}{{\phi }_{j_0}^{\prime }\left(t\right)}}\mathrm{d}(sin\left(A\left(t\right)\right))+\underset{T}{\overset{2T}{\int }}o\left(1\right)cos\left(A\left(t\right)\right)\mathrm{d}t.\hfill \end{array}$$

(5)

The function ${\phi }_{j_0}^{\prime }\left(t\right)$, by $2^{\circ }$ of the class $U_r\left(T_0\right)$, is monotonic and non-negative. Therefore, by the second mean value theorem,

$$\begin{array}{c}\hfill \underset{T}{\overset{2T}{\int }}{\displaystyle \frac{1}{{\phi }_{j_0}^{\prime }\left(t\right)}}\mathrm{d}(sin\left(A\left(t\right)\right))=\left\{\begin{array}{cc}{\displaystyle \frac{1}{{\phi }_{j_0}^{\prime }\left(T\right)}}\underset{T}{\overset{\xi }{\int }}\mathrm{d}(sin\left(A\left(t\right)\right))\hfill & \mathrm{if}{\phi }_{j_0}^{\prime }\left(t\right)\mathrm{is}\mathrm{increasing},\hfill \\ {\displaystyle \frac{1}{{\phi }_{j_0}^{\prime }\left(2T\right)}}\underset{\xi }{\overset{2T}{\int }}\mathrm{d}(sin\left(A\left(t\right)\right))\hfill & \mathrm{if}{\phi }_{j_0}^{\prime }\left(t\right)\mathrm{is}\mathrm{decreasing},\hfill \end{array}\right.\end{array}$$

$T⩽\xi ⩽2T$. Since ${\phi }_{j_0}^{\prime }\left(T\right)⩾\frac{{\phi }_{j_0}\left(T\right)}{T}$ and ${\phi }_{j_0}\left(T\right)\to \infty$ as $T\to \infty$, we have that

$$\underset{T}{\overset{2T}{\int }}{\displaystyle \frac{1}{{\phi }_{j_0}^{\prime }\left(t\right)}}\mathrm{d}(sin\left(A\left(t\right)\right))=o\left(T\right),T\to \infty .$$

This and (5) show that

$$\underset{T}{\overset{2T}{\int }}cos\left(A\left(t\right)\right)\mathrm{d}t=o\left(T\right),T\to \infty .$$

Similarly, it follows that

$$\underset{T}{\overset{2T}{\int }}sin\left(A\left(t\right)\right)\mathrm{d}t=o\left(T\right),T\to \infty .$$

Thus, in view of (3), in the case $({\underline{k}}_1,\dots ,{\underline{k}}_r)=(\underline{0},\dots ,\underline{0})$,

$$\underset{T\to \infty }{lim}{f}_T({\underline{k}}_1,\dots ,{\underline{k}}_r)=0,$$

and this together with (4) proves the lemma. □

3. Case of Absolute Convergence

Lemma 1 and the properties of weak convergence make it possible to obtain a limit lemma for ${\underline{\zeta }}_N(\underline{\sigma };\underline{Q})$ involving certain absolutely convergent Dirichlet series. Let $\theta >0$ be a fixed number, and

$$v_N\left(m\right)=exp\left\{-{\left({\displaystyle \frac{m}{N}}\right)}^{\theta }\right\},m,N\in \mathbb{N}.$$

Define

$$L_N\left({\sigma }_j-{\displaystyle \frac{n_j}{2}}+1,{\psi }_{lj}\right)=\sum _{m=1}^{\infty }{\displaystyle \frac{{\psi }_{lj}\left(m\right)v_N\left(m\right)}{m^{{\sigma }_j-\frac{n_j}{2}+1}}},l=1,\dots ,L_j,j=1,\dots ,r.$$

Since $v_N\left(m\right)$ with respect to m decreases exponentially, the latter series are absolutely convergent for all finite ${\sigma }_j$. Moreover, as $n_j⩾4$, we have that ${\sigma }_j>\frac{n_j-1}{2}⩾\frac{3}{2}>1$. Hence, the series for $L({\sigma }_j,{\chi }_{kj})$, $k=1,\dots ,K_j$, $j=1,\dots ,r$, are absolutely convergent. Therefore,

$${\zeta }_N({\sigma }_j;Q_j)=\sum _{k=1}^{K_j}\sum _{l=1}^{L_j}{\displaystyle \frac{a_{klj}}{k^{{\sigma }_j}{l}^{{\sigma }_j}}}L({\sigma }_j,{\chi }_{kj})L_N\left({\sigma }_j-{\displaystyle \frac{n_j}{2}}+1,{\psi }_{lj}\right)+\sum _{m=1}^{\infty }{\displaystyle \frac{f_{Q_j}\left(m\right)}{m^{{\sigma }_j}}},$$

(6)

where $j=1,\dots ,r$, is a combination of absolutely convergent Dirichlet series. Let

$${\underline{\zeta }}_N(\underline{\sigma };\underline{Q})=\left({\zeta }_N({\sigma }_1;Q_1),\dots ,{\zeta }_N({\sigma }_r;Q_r)\right),$$

and

$${\underline{\zeta }}_N(\underline{\sigma },\omega ;\underline{Q})=\left({\zeta }_N({\sigma }_1,{\omega }_1;Q_1),\dots ,{\zeta }_N({\sigma }_r,{\omega }_r;Q_r)\right),$$

where ${\zeta }_N({\sigma }_j,{\omega }_j;Q_j)$ is obtained from $\zeta ({\sigma }_j,{\omega }_j;Q_j)$ by putting $L_N\left({\sigma }_j-\frac{n_j}{2}+1,{\psi }_{lj}\right)$ in the place of $L\left({\sigma }_j-\frac{n_j}{2}+1,{\psi }_{lj}\right)$, where $j=1,\dots ,r$. Then, ${\zeta }_N({\sigma }_j,{\omega }_j;Q_j)$, where $j=1,\dots ,r$, is also a combination of absolutely convergent Dirichlet series. Let the function $u_{N,\underline{\sigma };\underline{Q}}:{\mathsf{\Omega }}^r\to {\mathbb{C}}^r$ be given by

$$u_N\left(\omega \right)=u_{N,\underline{\sigma };\underline{Q}}\left(\omega \right)={\underline{\zeta }}_N(\underline{\sigma },\omega ;\underline{Q}),{\sigma }_j>{\displaystyle \frac{n_j-1}{2}},j=1,\dots ,r.$$

In virtue of the absolute convergence of the series in ${\underline{\zeta }}_N(\underline{\sigma },\omega ;\underline{Q})$, the function $u_N\left(\omega \right)$ is continuous, and hence $(\mathcal{B}\left({\mathsf{\Omega }}^r\right),\mathcal{B}\left({\mathbb{C}}^r\right))$-measurable. Then, the measure $W_N=W_{N,\underline{\sigma };\underline{Q}}=m_H{u}_N^{-1}$, where

$$m_H{u}_N^{-1}\left(A\right)=m_H\left(u_N^{-1}A\right),A\in \mathcal{B}\left({\mathbb{C}}^r\right),$$

can be defined. For $A\in \mathcal{B}\left({\mathbb{C}}^r\right)$, define

$$P_{T,N}\left(A\right)=P_{T,N,\underline{\sigma };\underline{Q}}\left(A\right)={\displaystyle \frac{1}{T}}\mathrm{meas}\left\{t\in [T,2T]:{\underline{\zeta }}_N(\underline{\sigma }+i\underline{\phi }\left(t\right);\underline{Q})\in A\right\}.$$

Lemma 2.

Suppose that $\left({\phi }_1\left(t\right),\dots ,{\phi }_r\left(t\right)\right)\in U_r\left(T_0\right)$. Then, $P_{T,N}$ converges weakly to $W_N$ as $T\to \infty$.

Proof. 

From the definitions of $R_T$, $P_{T,N}$ and $u_N$, we have

$$\begin{array}{ccc}\hfill P_{T,N}\left(A\right)& =& {\displaystyle \frac{1}{T}}\mathrm{meas}\left\{t\in [T,2T]:u_N\left(\left(p^{-i{\phi }_1\left(t\right)}:p\in \mathbb{P}\right),\dots ,\left(p^{-i{\phi }_r\left(t\right)}:p\in \mathbb{P}\right)\right)\in A\right\}\hfill \\ & =& {\displaystyle \frac{1}{T}}\mathrm{meas}\left\{t\in [T,2T]:\left(\left(p^{-i{\phi }_1\left(t\right)}:p\in \mathbb{P}\right),\dots ,\left(p^{-i{\phi }_r\left(t\right)}:p\in \mathbb{P}\right)\right)\in u_N^{-1}A\right\}\hfill \\ & =& R_T\left(u_N^{-1}A\right)\hfill \end{array}$$

for all $A\in \mathcal{B}\left({\mathbb{C}}^r\right)$. Therefore, $P_{T,N}=R_T{u}_N^{-1}$. Now, Lemma 1 and the preservation of weak convergence under continuous mappings, see, for example, Theorem 5.1 of [24], show that $P_{T,N}$ converges weakly to the probability measure $m_H{u}_N^{-1}$ as $T\to \infty$. □

The measure $W_N$ is an important ingredient of the proof of Theorem 3. We see that $W_N$ is independent on the functions ${\phi }_1\left(t\right),\dots ,{\phi }_r\left(t\right)$. Therefore, we can use some statements from [22] to prove the following lemma.

Lemma 3.

The probability measure $W_N$ converges weakly to $P_{\underline{\zeta }}$ as $N\to \infty$.

Proof. 

By Lemma 8 from [22], the sequence of probability measures $\{W_N:N\in \mathbb{N}\}$ is tight, i.e., for every $ϵ>0$, there exists a compact set $K=K\left(ϵ\right)\in {\mathbb{C}}^r$ such that

$$W_N\left(K\right)>1-ϵ$$

for all $N\in \mathbb{N}$. Hence, by the Prokhorov theorem, see, for example, Theorem 6.1 of [24], the above sequence is relatively compact. This means that every subsequence of $\left\{W_N\right\}$ has a subsequence weakly convergent to a certain probability measure on $({\mathbb{C}}^r,\mathcal{B}\left({\mathbb{C}}^r\right))$. Thus, there exists a sequence $\left\{W_{N_l}\right\}\subset \left\{W_N\right\}$ such that $W_{N_l}$ converges weakly to the measure $P_{\underline{\sigma };\underline{Q}}$ as $l\to \infty$. In the proof of Theorem 2 from [22], it is obtained that $P_{\underline{\sigma };\underline{Q}}$ coincides with $P_{\underline{\zeta }}$. Since the sequence $\left\{W_N\right\}$ is relatively compact, from this we have that $W_N$ converges weakly to $P_{\underline{\zeta }}$ as $N\to \infty$. □

4. Estimate in the Mean

To derive Theorem 3 from Lemma 2, we have to show the nearest ${\underline{\zeta }}_N(\underline{\sigma }+i\underline{\phi }\left(t\right);\underline{Q})$ to $\underline{\zeta }(\underline{\sigma }+i\underline{\phi }\left(t\right);\underline{Q})$. Let, for ${\underline{s}}_j=(s_{j1},\dots ,s_{jr})$, $j=1,2$,

$$d_r({\underline{s}}_1,{\underline{s}}_2)={\left(\sum _{j=1}^r|s_{j1}-s_{j2}|\right)}^{1/2}.$$

Lemma 4.

Suppose that $\left({\phi }_1\left(t\right),\dots ,{\phi }_r\left(t\right)\right)\in U_r\left(T_0\right)$ and ${\sigma }_j>\frac{n_j-1}{2}$, $j=1,\dots ,r$. Then,

$$\underset{N\to \infty }{lim}\underset{T\to \infty }{lim\; sup}{\displaystyle \frac{1}{T}}\underset{T}{\overset{2T}{\int }}{d}_r\left(\underline{\zeta }(\underline{\sigma }+i\underline{\phi }\left(t\right);\underline{Q}),{\underline{\zeta }}_N(\underline{\sigma }+i\underline{\phi }\left(t\right);\underline{Q})\right)\mathrm{d}t=0.$$

Proof. 

Clearly,

$$\begin{array}{ccc}& & \underset{T}{\overset{2T}{\int }}{d}_r\left(\underline{\zeta }(\underline{\sigma }+i\underline{\phi }\left(t\right);\underline{Q}),{\underline{\zeta }}_N(\underline{\sigma }+i\underline{\phi }\left(t\right);\underline{Q})\right)\mathrm{d}t\hfill \\ \hfill \ll & & \sum _{j=1}^r\underset{T}{\overset{2T}{\int }}|\zeta ({\sigma }_j+i{\phi }_j\left(t\right);Q_j)-{\zeta }_N({\sigma }_j+i{\phi }_j\left(t\right);Q_j)|\mathrm{d}t.\hfill \end{array}$$

Therefore, it suffices to show that

$$\underset{N\to \infty }{lim}\underset{T\to \infty }{lim\; sup}{\displaystyle \frac{1}{T}}\underset{T}{\overset{2T}{\int }}\left|\zeta (\sigma +i\phi \left(t\right);Q)-{\zeta }_N(\sigma +i\phi \left(t\right);Q)\right|\mathrm{d}t=0$$

for Q and $\sigma$ satisfying the hypotheses of Theorem 1, and $\phi \left(t\right)$ satisfying $1^{\circ }$ and $2^{\circ }$ of the class $U_r\left(T_0\right)$. However, the latter equality was proved in [23] and Lemma 2. We only mention that, in view of (6), it is suficient to prove that

$$\underset{N\to \infty }{lim}\underset{T\to \infty }{lim\; sup}{\displaystyle \frac{1}{T}}\underset{T}{\overset{2T}{\int }}\left|L(\sigma +i\phi \left(t\right),\chi )-L_N(\sigma +i\phi \left(t\right),\chi )\right|\mathrm{d}t=0$$

(7)

for $\sigma >\frac{1}{2}$. For this, the representation

$$L_N(s,\chi )={\displaystyle \frac{1}{2\pi i}}\underset{\theta -i\infty }{\overset{\theta +i\infty }{\int }}L(s+z,\chi )l_N\left(z\right){\displaystyle \frac{\mathrm{d}z}{z}}$$

with $l_N\left(z\right)=\frac{z}{\theta }\mathsf{\Gamma }\left(\frac{z}{\theta }\right)N^z$, is applied. Thus, the latter representation, the mean square estimate

$$\underset{T}{\overset{2T}{\int }}{\left|L(\sigma +i\phi (t)+i\tau ,\chi )\right|}^2\mathrm{d}t{\ll }_{\sigma ,\chi ,\phi }T\left(1+\left|\tau \right|\right)$$

for all real $\tau$, and the classical estimate

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

lead to equality (7). □

5. Proof of Theorem 3

Theorem 3 follows from Lemmas 2–4 and the following statement on convergence in distribution $\left(\stackrel{\mathcal{D}}{\to }\right)$; see, for example, Theorem 4.2 from [24].

Lemma 5.

Suppose that the space $(\mathbb{X},\rho )$ is separable, and the $\mathbb{X}$-valued random elements $X_{nk}$ and $Y_n$, $n,k\in \mathbb{N}$, are defined on the same probability space with measure P. Let

$$X_{nk}\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}{X}_{k}and{X}_k\underset{k\to \infty }{\overset{\mathcal{D}}{\to }}{X}_{,}$$

and, for every$ϵ>0$,

$$\underset{k\to \infty }{lim}\underset{n\to \infty }{lim\; sup}P\left\{\rho \left(X_{nk},Y_n\right)⩾ϵ\right\}=0.$$

Then,$Y_n\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}X$.

Proof of Theorem 3 

Let ${\xi }_T$ be a random variable defined on a certain probability space with the measure P and uniformly distributed in the interval $[T,2T]$. Define the ${\mathbb{C}}^r$-valued random element

$$X_{T,N}=X_{T,N,\underline{\sigma };\underline{Q}}={\underline{\zeta }}_N(\underline{\sigma }+i\underline{\phi }\left({\xi }_T\right);\underline{Q})$$

and

$$X_T=X_{T,\underline{\sigma };\underline{Q}}=\underline{\zeta }(\underline{\sigma }+i\underline{\phi }\left({\xi }_T\right);\underline{Q}),$$

and denote by $Y_N=Y_{N,\underline{\sigma };\underline{Q}}$ the ${\mathbb{C}}^r$-valued random element having the distribution $W_N$. Then, in view of Lemma 2, we have

$$X_{T,N}\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{X}_N,$$

(8)

while Lemma 3 implies that

$$Y_N\underset{N\to \infty }{\overset{\mathcal{D}}{\to }}{P}_{\underline{\zeta }}.$$

(9)

Moreover, by Lemma 4, for every $ϵ>0$,

$$\begin{array}{ccc}& & \underset{N\to \infty }{lim}\underset{T\to \infty }{lim\; sup}P\left\{d_r\left(X_{T,N},X_T\right)⩾ϵ\right\}\hfill \\ \hfill =& & \underset{N\to \infty }{lim}\underset{T\to \infty }{lim\; sup}{\displaystyle \frac{1}{T}}\mathrm{meas}\left\{t\in [T,2T]:d_r\left(\underline{\zeta }(\underline{\sigma }+i\underline{\phi }\left(t\right);\underline{Q}),{\underline{\zeta }}_N(\underline{\sigma }+i\underline{\phi }\left(t\right);\underline{Q})\right)⩾ϵ\right\}\hfill \\ \hfill ⩽& & \underset{N\to \infty }{lim}\underset{T\to \infty }{lim\; sup}{\displaystyle \frac{1}{ϵT}}\underset{T}{\overset{2T}{\int }}{d}_r\left(\underline{\zeta }(\underline{\sigma }+i\underline{\phi }\left(t\right);\underline{Q}),{\underline{\zeta }}_N(\underline{\sigma }+i\underline{\phi }\left(t\right);\underline{Q})\right)\mathrm{d}t=0.\hfill \end{array}$$

This, together with (8) and (9), shows that the random elements $X_{T,N}$, $X_T$ and $Y_N$ satisfy the hypotheses of Lemma 5. Therefore,

$$X_T\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{P}_{\underline{\zeta }},$$

and this is equivalent to the assertion of the theorem. □

6. Conclusions

For $j=1,\dots ,r$, let $Q_j$ be a positive definite $n_j\times n_j$ matrix, such that ${\underline{x}}^{\mathrm{T}}{Q}_j\underline{x}\in \mathbb{Z}$ for all $\underline{x}\in {\mathbb{Z}}^r\setminus \left\{\underline{0}\right\}$, $n_j\in 2\mathbb{N}$ and $n_j⩾4$. In this paper, it is obtained that, for a collection of Epstein zeta-functions $\underline{\zeta }(\underline{s};\underline{Q})=\left(\zeta (s_1;Q_1),\dots ,\zeta (s_r;Q_r)\right)$, a limit theorem on weakly convergent probability measures with generalized shifts $\underline{\zeta }(\underline{\sigma }+i\underline{\phi }\left(t\right);\underline{Q})$ is valid, where ${\phi }_j\left(t\right)$ are certain differentiable functions. The proven theorem generalizes the main result of [22] obtained for ${\phi }_j\left(t\right)=t$. Note that the main theorem remains valid even if the functions ${\phi }_j\left(t\right)$ coincide for some j. For example, one may consider ${\phi }_j\left(t\right)=t{log}^{{\alpha }_j}t$ with different ${\alpha }_j>0$. Also, polynomials ${\phi }_j\left(t\right)=a_j{t}^{{\alpha }_j}+\dots +a_0$, with $a_j>0$ and different ${\alpha }_j>0$, can be used.

As shown in the proof of Lemma 1, using generalized shifts $\zeta ({\sigma }_j+i{\phi }_j\left(t\right);Q_j)$ makes it possible to obtain a desired rate of convergence for $R_T$ to $m_H$. We conjecture that this phenomenon is also preserved for the measures $P_T$ and $P_{\underline{\zeta }}$.

The next paper will be devoted to a joint generalized discrete version, i.e., for weak convergence of

$${\displaystyle \frac{1}{N+1}}\#\left\{N⩽k⩽2N:\underline{\zeta }(\underline{\sigma }+i\underline{\phi }\left(k\right);\underline{Q})\in A\right\},A\in \mathcal{B}\left({\mathbb{C}}^r\right),$$

as $N\to \infty$. Here, $\#A$ denotes the number of elements of the set A.