Joint Approximation of Analytic Functions by Shifts of Hurwitz Zeta-Functions in Short Intervals

Abstract

For $j=1,\dots ,r$, ${\alpha }_j\in (0,1)$ are such that the set $\{\log (m+{\alpha }_j):m\in {\mathbb{N}}_0,j=1,\dots ,r\}$ is linearly independent over $\mathbb{Q}$. In the paper, we consider simultaneous approximation of a tuple of analytic functions by shifts $(\zeta (s+i\tau ,{\alpha }_1),\dots ,\zeta (s+i\tau ,{\alpha }_r))$ of the Hurwitz zeta-functions. We prove that the set of these shifts has a positive lower density in the interval $[T,T+H]$ for $T^{23/70}⩽H⩽T$. This extends the known results obtained for algebraically independent ${\alpha }_j$, $j=1,\dots ,r$, and with $T^{27/82}⩽H⩽T^{1/2}$. Moreover, we obtain a theorem on approximation of analytic functions by a linear combination of derivatives of Hurwitz zeta-functions. From that theorem, it follows that the number of zeros of that linear combination lying in the region $\{s\in \mathbb{C}:1/2<{\sigma }_1<{\sigma }_2<1,t\in [T,T+H]\}$ is greater than $cH$ with some $c>0$. This generalizes and extends the classical Voronin result for the function $\zeta (s,\alpha )$ with rational $\alpha$. For the proofs, a probabilistic approach is used.

1. Introduction

Let $\alpha$, $0<\alpha ⩽1$, be a fixed parameter. The Hurwitz zeta-function $\zeta (s,\alpha )$, $s=\sigma +it$, is one of the analytic functions that are widely used in number theory. It was introduced by A. Hurwitz in [1], and, for $\sigma >1$, is defined by the Dirichlet series:

$$\zeta (s,\alpha )={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{1}{{(m+\alpha )}^s}}.$$

Moreover, the function $\zeta (s,\alpha )$ has analytic continuation to the whole complex plane, except for the point $s=1$ which is its simple pole with residue 1. The Hurwitz zeta-function covers the Riemann zeta-function

$$\zeta \left(s\right)={\displaystyle \sum _{m=1}^{\infty }}{\displaystyle \frac{1}{m^s}},$$

namely, $\zeta (s,1)=\zeta \left(s\right)$, and

$$\zeta \left(s,\frac{1}{2}\right)=\left(2^s-1\right)\zeta \left(s\right).$$

Analytic properties of the Hurwitz zeta-function depend on arithmetic of the parameter $\alpha$, and differ significantly from those of $\zeta \left(s\right)$. From the Euler product over primes

$$\zeta \left(s\right)=\prod _p{\left(1-\frac{1}{p^s}\right)}^{-1},\sigma >1,$$

it follows easily that $\zeta \left(s\right)$ has no zeros in the half-plane $\sigma >1$, while $\zeta (s,\alpha )$, for $\alpha \ne 1$ or $\alpha \ne 1/2$, possesses zeros lying in that half-plane. In the case of rational or transcendental $\alpha$, this has been proved by H. Davenport and H. Heilbronn in [2], while this has been done for algebraic irrational $\alpha$ by J.W.S. Cassels in [3].

The function $\zeta (s,\alpha )$ is an important object of analytic number theory; however, it is also useful in special function theory, mathematical statistics and even in quantum physics. The classical theory of $\zeta (s,\alpha )$ is presented in [4,5]; see also [6,7].

In this paper, we are interested in approximation properties of Hurwitz zeta-functions and their applications for zero-distribution.

The function $\zeta (s,\alpha )$, as with some other zeta- and L-functions, for some classes of the parameter $\alpha$, is universal in the sense that its shifts, $\zeta (s+i\tau )$, $\tau \in \mathbb{R}$, approximate an entire class of analytic functions. S.M. Voronin is the discoverer of universality for $\zeta \left(s\right)$. Let $0<r<1/4$, $K=\{s\in \mathbb{C}:|s|⩽r\}$ and $f\left(s\right)$ be a continuous non-vanishing function on K which is analytic inside of K. Then, Voronin proved in [8]—see also [9,10,11,12,13]—that, for every $\epsilon >0$, there exists $\tau =\tau \left(\epsilon \right)\in \mathbb{R}$ such that

$$\underset{|s|⩽r}{\max }\left|\zeta \left(s+\frac{3}{4}+i\tau \right)-f\left(s\right)\right|<\epsilon .$$

Moreover, he obtained joint universality of Dirichlet L-functions $L(s,{\chi }_1),\dots ,L(s,{\chi }_n)$ with pairwise non-equivalent characters ${\chi }_1,\dots ,{\chi }_r$ [10], and, from this, derived [11] the lower bound for the number of zeros of the function $\zeta (s,\alpha )$ with rational $\alpha =a/q$, $(a,q)=1$, $0<a<q$, $q\ne 1/2$. More precisely, he proved that, for arbitrary ${\sigma }_1$, ${\sigma }_2$, $1/2<{\sigma }_1<{\sigma }_2<1$, there is a constant $c=c(\alpha ,{\sigma }_1,{\sigma }_2)>0$ such that, for sufficiently large T, the function $\zeta (s,\alpha )$ has more than $cT$ zeros in the region $\{s\in \mathbb{C}:{\sigma }_1<\sigma <{\sigma }_2,|t|⩽T\}$. A proof of this result in non-explicit form contains universality of $\zeta (s,\alpha )$ with rational $\alpha$. A full proof of universality of $\zeta (s,\alpha )$ with rational $\alpha$ has been given in [14,15]. We state below a modern version of universality for the function $\zeta (s,\alpha )$. We use the following notation. Let $\mathcal{K}$ be the class of compact subsets of the strip $\mathcal{D}=\{s\in \mathbb{C}:1/2<\sigma <1\}$ with connected complement, let $H\left(K\right)$ with $K\in \mathcal{K}$ be the class of continuous functions on K that are analytic inside of K, and let $\mathrm{meas}A$ denote the Lebesgue measure of a measurable set $A\subset \mathbb{R}$.

Proposition 1

(See [14,15,16,17,18,19,20,21,22,23]). Suppose that the parameter α is transcendental, or rational $\alpha \ne 1$ or $\alpha \ne 1/2$. Then, for every $K\in \mathcal{K}$, $f\left(s\right)\in H\left(K\right)$ and $\epsilon >0$,

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

Moreover, the limit

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

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

In [23,24], it was observed that the transcendence of $\alpha$ in Proposition 1 can be replaced by a more general requirement on the linear independence over the field of rational numbers $\mathbb{Q}$ for the set $\{\log (m+\alpha ):m\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}\}$.

The case of algebraic irrational $\alpha$ is the most complicated one, and the effective result in that case is given in [25].

Proposition 2

(See Theorem 2 of [25]). Let α be an algebraic irrational number of degree d, $d_1=4{\left(27{\left(4.45\right)}^2\right)}^{-1}$, $d_2=d_1/d^2$, ${\epsilon }_1\in (0,d_2)$, $1-d_2+{\epsilon }_1⩽{\sigma }_0⩽1$ and $s_0={\sigma }_0+i{t}_0$. Suppose $f\left(s\right)$ is a function continuous and analytic in the interior of the disc $|s-s_0|⩽r$, $r>0$, $0<a<1$ and ${\epsilon }_2\in (0,|f\left(s_0\right)|)$. Then, for all but finitely many $\alpha \in [a,1]$ of degree at most $d_0-2d_1/d_0+{\epsilon }_1$ with

$$d_0⩽{\displaystyle \frac{d_1}{1-{\sigma }_0+{\epsilon }_1}},$$

there is $\tau \in [T,2T]$ and $\delta =\delta ({\epsilon }_2,f,T)>0$ such that

$$\underset{|s-s_0|⩽\delta r}{\max }|\zeta (s+i\tau ,\alpha )-f\left(s\right)|⩽3{\epsilon }_2,$$

where $T=T({\epsilon }_2,f,\alpha )$ is explicitly given. The set of exceptional α is described effectively, and δ is effectively computable as well.

Proposition 2 illustrates the importance of the degree of $\alpha$ in the universality problem.

Proposition 1 has an extension for the case of the so-called short intervals when density of approximating shifts is investigated in the interval $[T,T+H]$ with $H/T\to 0$ as $T\to \infty$.

For brevity, denote

$$U_{T,H}(\dots )={\displaystyle \frac{1}{H}}\mathrm{meas}\{\tau \in [T,T+H]:\dots \},$$

and in place of dots write a condition that $\tau$ satisfies.

Proposition 3

(See [26,27]). Let the parameter α be transcendental or rational $\alpha =a/q$, $(a,q)=1$, $a<q$, $q>2$, and $T^{23/70}⩽H⩽T$. Then, for every $K\in \mathcal{K}$, $f\left(s\right)\in H\left(K\right)$ and $\epsilon >0$,

$$\underset{T\to \infty }{\mathrm{lim\; inf}} U_{T,H}\left(\underset{s\in K}{\sup }|\zeta (s+i\tau ,\alpha )-f\left(s\right)|<\epsilon \right)>0.$$

Moreover, for every $K\in \mathcal{K}$ and $f\left(s\right)\in H\left(K\right)$, the limit

$$\underset{T\to \infty }{\lim }{U}_{T,H}\left(\underset{s\in K}{\sup }|\zeta (s+i\tau ,\alpha )-f\left(s\right)|<\epsilon \right)$$

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

The case of transcendental $\alpha$ was obtained in [26], while the rational case is treated in [27].

For the proof of Proposition 3, the important role belongs to the new mean square estimate [26]

$${\displaystyle \underset{T-H}{\overset{T+H}{\int }}}{|\zeta (\sigma +it,\alpha )|}^2\mathrm{d}t{\ll }_{\alpha ,\sigma }H$$

(1)

which is uniform in H, $T^{23/70}⩽H⩽T$, and is valid for $1/2<\sigma ⩽31/52$.

In the case of $\alpha =1$ or $\alpha =1/2$, more precise results are known. Denote by $H_0\left(K\right)$ a subclass of $H\left(K\right)$, $K\in \mathcal{K}$, of functions non-vanishing on K. Let $\eta =1273/4051$. Observe that $\eta <23/70$.

Proposition 4

(See [27,28]). Suppose that $\alpha =1$ or $\alpha =1/2$, and $T^{\eta }⩽H⩽T$. Then, for every $K\in \mathcal{K}$, $f\left(s\right)\in H_0\left(K\right)$ and $\epsilon >0$, the statements of Proposition 3 are valid.

The proof of the proposition is based on the estimate obtained in [28]

$${\displaystyle \underset{T-H}{\overset{T+H}{\int }}}{|\zeta (\sigma +it)|}^2\mathrm{d}t{\ll }_{\sigma }H$$

for some $\sigma >1/2$ and $T^{\eta }⩽H⩽T$.

Proposition 1 allows obtaining certain information on the number of zeros of $\zeta (s,\alpha )$ lying in the strip $\mathcal{D}$. The following statement is known [11].

Proposition 5

(See [11]). Suppose that $\alpha =a/q$, $(a,q)=1$, $1<a<q$, $\alpha \ne 1/2$. Then, for every ${\sigma }_1,{\sigma }_2$, $1/2<{\sigma }_1<{\sigma }_2<1$, there exists $c=c(\alpha ,{\sigma }_1,{\sigma }_2)>0$ such that, for sufficiently large T, the function $\zeta (s,\alpha )$ has more than $cT$ zeros in the region $\{s\in \mathbb{C}:{\sigma }_1<\sigma <{\sigma }_2,|t|<T\}$.

Using Proposition 3 extends Proposition 5 for short intervals.

Proposition 6

(See Theorem 4 of [27]). Suppose that the parameter α is transcendental or rational $\ne 1$ or $\ne 1/2$, and $T^{23/70}⩽H⩽T$. Then, for every ${\sigma }_1,{\sigma }_2$, $1/2<{\sigma }_1<{\sigma }_2<1$, there is a constant $c=c(\alpha ,{\sigma }_1,{\sigma }_2)>0$ such that, for sufficiently large T, the function $\zeta (s,\alpha )$ has more than $cH$ zeros in the region $\{s\in \mathbb{C}:{\sigma }_1<\sigma <{\sigma }_2,T⩽t⩽T+H\}$.

Universality of zeta-functions has a joint generalization, i.e., a simultaneous approximation of a tuple of analytic functions $(f_1\left(s\right),\dots ,f_r\left(s\right))$ by a tuple of considered zeta-functions. For Hurwitz zeta-functions, the following result is known.

Proposition 7

(See Theorem 1 of [24]). Suppose that, for $j=1,\dots ,r$, ${\alpha }_j\in (0,1)$, and the set

$$\mathcal{L}({\alpha }_1,\dots ,{\alpha }_r)\stackrel{\mathrm{def}}{=}\left\{\log (m+{\alpha }_j),j=1,\dots ,r,m\in {\mathbb{N}}_0\right\}$$

is linearly independent over $\mathbb{Q}$. For $j=1,\dots ,r$, let $K_j\in \mathcal{K}$ and $f_j\left(s\right)\in H\left(K_j\right)$. Then, for every $\epsilon >0$,

$$\underset{T\to \infty }{\mathrm{lim\; inf}} {\displaystyle \frac{1}{T}}\mathrm{meas}\left\{\tau \in [0,T]:\underset{1⩽j⩽r}{\sup }\underset{s\in K_j}{\sup }\left|\zeta (s+i\tau ,{\alpha }_j)-f_j\left(s\right)\right|<\epsilon \right\}>0.$$

We recall that the numbers $a_1,\dots ,a_r$ are said to be algebraically independent over $\mathbb{Q}$ if there does not exist a polynomial $p(s_1,\dots ,s_r)\cancel{\equiv }0$ with rational coefficients such that $p(a_1,\dots ,a_r)=0$. It is not difficult to see that if the numbers ${\alpha }_1,\dots ,{\alpha }_r$ are algebraically independent over $\mathbb{Q}$, then the set $L({\alpha }_1,\dots ,{\alpha }_r)$ (it can be a multiset) is linearly independent over $\mathbb{Q}$.

The purpose of this article is a certain effectivization of joint universality for Hurwitz zeta-functions. More precisely, we extend Proposition 7 for short intervals. The first attempt in this case was made in [29] for algebraically independent ${\alpha }_1,\dots ,{\alpha }_r$ and $T^{27/82}⩽H⩽T^{1/2}$. The purpose of this paper is to substitute the algebraic independence of ${\alpha }_1,\dots ,{\alpha }_r$ for the linear independence over $\mathbb{Q}$ of the set $\mathcal{L}({\alpha }_1,\dots ,{\alpha }_r)$ as well as to extend the interval for H until $T^{23/70}⩽H⩽T$. Observe that $23/70<27/82$. We notice that the linear independence of $\mathcal{L}({\alpha }_1,\dots ,{\alpha }_r)$ is a more general requirement than the algebraic independence of ${\alpha }_1,\dots ,{\alpha }_r$. Unfortunately, concrete examples of such algebraically dependent numbers ${\alpha }_1,\dots ,{\alpha }_r$ for which the set $\mathcal{L}({\alpha }_1,\dots ,{\alpha }_r)$ is linearly independent over $\mathbb{Q}$ and ${\alpha }_1,\dots ,{\alpha }_r$ are not known. An extension of the interval for H is closely related to new estimates for the mean square in short intervals. Moreover, we obtain universality of a linear combination of derivatives of Hurwitz zeta-functions, and the lower estimate for a number of zeros of that combination in short intervals.

For the proofs, we develop the probabilistic approach which reflects the results of previous works. Our innovation relies on the new mean square estimate in short intervals obtained in [26].

In the next section, we state the results of the paper. Section 3 contains a probabilistic background. Section 4, Section 5 and Section 6 are devoted to proofs of theorems stated in Section 2.

2. Statements of Results

Theorem 1.

Suppose that, for $j=1,\dots ,r$, ${\alpha }_j\in (0,1)$, the set $\mathcal{L}({\alpha }_1,\dots ,{\alpha }_r)$ is linearly independent over $\mathbb{Q}$, and $T^{23/70}⩽H⩽T$. For $j=1,\dots ,r$, let $K_j\in \mathcal{K}$ and $f_j\left(s\right)\in H\left(K_j\right)$. Then, for any $\epsilon >0$,

$$\underset{T\to \infty }{\mathrm{lim\; inf}} U_{T,H}\left(\underset{1⩽j⩽r}{\sup }\underset{s\in K_j}{\sup }\left|\zeta (s+i\tau ,{\alpha }_j)-f_j\left(s\right)\right|<\epsilon \right)>0.$$

Moreover, for every $K_j\in \mathcal{K}$ and $f_j\left(s\right)\in H\left(K_j\right)$, $j=1,\dots ,r$, the limit

$$\underset{T\to \infty }{\lim }{U}_{T,H}\left(\underset{1⩽j⩽r}{\sup }\underset{s\in K_j}{\sup }\left|\zeta (s+i\tau ,{\alpha }_j)-f_j\left(s\right)\right|<\epsilon \right)$$

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

By $\mathcal{H}\left(\mathcal{D}\right)$ denote the space of analytic functions on $\mathcal{D}$ equipped with the topology of uniform convergence on compacta. For $g_1\left(s\right),\dots ,g_r\left(s\right)$ define $u:{\mathcal{H}}^r\left(\mathcal{D}\right)\to \mathcal{H}\left(\mathcal{D}\right)$ by

$$u(g_1\left(s\right),\dots ,g_r\left(s\right))={\displaystyle \sum _{j=1}^r}{c}_j{g}_j^{\left(m_j\right)}\left(s\right),$$

where, for $j=1,\dots ,r$, $c_j\in \mathbb{C}$ are not all zero, and $m_j\in {\mathbb{N}}_0$.

Theorem 2.

Suppose that, for ${\alpha }_j\in (0,1)$, $j=1,\dots ,r$, the set $\mathcal{L}({\alpha }_1,\dots ,{\alpha }_r)$ is linearly independent over $\mathbb{Q}$, and $T^{23/70}⩽H⩽T$. Let $K\in \mathcal{K}$ and $f\left(s\right)\in H\left(K\right)$. Then, for every $\epsilon >0$,

$$\underset{T\to \infty }{\mathrm{lim\; inf}} U_{T,H}\left(\underset{s\in K}{\sup }\left|u\left(\zeta (s+i\tau ,{\alpha }_1),\dots ,\zeta (s+i\tau ,{\alpha }_r)\right)-f\left(s\right)\right|<\epsilon \right)>0.$$

Moreover, for every $K\in \mathcal{K}$ and $f\left(s\right)\in H\left(K\right)$, the limit

$$\underset{T\to \infty }{\lim }{U}_{T,H}\left(\underset{s\in K}{\sup }\left|u\left(\zeta (s+i\tau ,{\alpha }_1),\dots ,\zeta (s+i\tau ,{\alpha }_r)\right)-f\left(s\right)\right|<\epsilon \right)$$

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

Theorem 3.

Suppose that, for ${\alpha }_j\in (0,1)$, $j=1,\dots ,r$, the set $\mathcal{L}({\alpha }_1,\dots ,{\alpha }_r)$ is linearly independent over $\mathbb{Q}$, and $T^{23/70}⩽H⩽T$. Then, for every ${\sigma }_1,{\sigma }_2$, $1/2<{\sigma }_1<{\sigma }_2<1$, there exists a constant $c=c({\alpha }_1,\dots ,{\alpha }_r,{\sigma }_1,{\sigma }_2)>0$ such that, for sufficiently large T, the function

$$u\left(\zeta (s,{\alpha }_1),\dots ,\zeta (s,{\alpha }_r)\right)$$

has more than $cH$ zeros in the rectangle $\{s\in \mathbb{C}:{\sigma }_1<\sigma <{\sigma }_2,T⩽t⩽T+H\}$.

3. Probabilistic Results

A problem of approximation by shifts of zeta-functions is closely connected to the statistical behaviour of zeta-functions; see [15,16,17]. In this section, we give a probabilistic background for joint distribution of Hurwitz zeta-functions in short intervals. In this approach, weak convergence of probability measures occupies a crucial role. Let $\mathcal{B}\left(\mathcal{Z}\right)$ denote the Borel $\sigma$-field of the space $\mathcal{Z}$ (topological in general), and $Q_n$, $n\in \mathbb{N}$, and Q be probability measures on $(\mathcal{Z},\mathcal{B}(\mathcal{Z}\left)\right)$. By the definition, $Q_n$ converges weakly to Q as $n\to \infty$, or shortly $Q_n{\displaystyle \underset{n\to \infty }{\overset{\mathrm{w}}{\to }}}Q$, if, for every real continuous bounded function g on $\mathcal{Z}$, the relation

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

holds. In this paper, we deal with the spaces $\mathcal{H}\left(\mathcal{D}\right)$ and ${\mathcal{H}}^r\left(\mathcal{D}\right)$. Recall that the space $\mathcal{H}\left(\mathcal{D}\right)$ is equipped with the topology of uniform convergence on compacta, and

$${\mathcal{H}}^r\left(\mathcal{D}\right)=\underset{r}{\underbrace{\mathcal{H}\left(\mathcal{D}\right)\times ···\times \mathcal{H}\left(\mathcal{D}\right)}}$$

uses the product topology.

Let, for brevity,

$$\underline{\zeta }(s,\underline{\alpha })=\left(\zeta (s,{\alpha }_1),\dots ,\zeta (s,{\alpha }_r)\right).$$

The main probabilistic object in this section is the probability measure

$$P_{T,H,\underline{\zeta }}\left(A\right)=U_{T,H}\left(\underline{\zeta }(s+i\tau ,\underline{\alpha })\in A\right),A\in \mathcal{B}\left({\mathcal{H}}^r\left(\mathcal{D}\right)\right),$$

and our aim is the weak convergence of $P_{T,H,\underline{\zeta }}$ as $T\to \infty$ to an explicitly given probability measure on $({\mathcal{H}}^r\left(\mathcal{D}\right),\mathcal{B}\left({\mathcal{H}}^r\left(\mathcal{D}\right)\right))$. The definition of the limit measure is based on one topological object. Define

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

with

$${\Omega }_j=\prod _{m\in {\mathbb{N}}_0}\left\{s\in \mathbb{C}:|s|=1\right\},j=1,\dots ,r.$$

In the set ${\Omega }_j$, the operation of pairwise multiplication and product topology can be introduced, and ${\Omega }_j$ becomes a compact topological group. Hence, ${\Omega }^r$ is a compact topological group. Therefore, on $({\Omega }^r,\mathcal{B}\left({\Omega }^r\right))$, the probability Haar measure $\mu$ exists, and this leads to the probability space $({\Omega }^r,\mathcal{B}\left({\Omega }^r\right),\mu )$. Denote elements of ${\Omega }^r$ by $\omega =({\omega }_1,\dots ,{\omega }_r)$, ${\omega }_j\in {\Omega }_j$, ${\omega }_j=({\omega }_j\left(m\right):m\in {\mathbb{N}}_0)$. Now, on the probability space $({\Omega }^r,\mathcal{B}\left({\Omega }^r\right),\mu )$, define the ${\mathcal{H}}^r\left(\mathcal{D}\right)$-valued random element

$$\underline{\zeta }\left(s,\underline{\alpha },\omega \right)=\left(\zeta (s,{\alpha }_1,{\omega }_1),\dots ,\zeta (s,{\alpha }_r,{\omega }_r)\right),$$

and let

$$P_{\underline{\zeta }}\left(A\right)=\mu \left(\underline{\zeta }(s,\underline{\alpha },\omega )\in A\right),A\in \mathcal{B}\left({\mathcal{H}}^r\left(\mathcal{D}\right)\right),$$

be its distribution.

The main result of this section is the following statement.

Theorem 4.

Suppose that, for $j=1,\dots ,r$, ${\alpha }_j\in (0,1)$, the set $\mathcal{L}({\alpha }_1,\dots ,{\alpha }_r)$ is linearly independent over $\mathbb{Q}$, and $T^{23/70}⩽H⩽T$. Then the relation

$$P_{T,H,\underline{\zeta }}{\displaystyle \underset{T\to \infty }{\overset{\mathrm{w}}{\to }}}{P}_{\underline{\zeta }}$$

holds.

3.1. Auxiliary Lemmas

We observe that the Haar measure $\mu$ on $({\Omega }^r,\mathcal{B}\left({\Omega }^r\right))$ is a convenient object which ensures a certain independence situation in probabilistic sense. We add that the Fourier transform of the Haar measure takes only two values: 0 or 1.

We begin with a limit lemma on the group ${\Omega }^r$. For $A\in \mathcal{B}\left({\Omega }^r\right)$, define

$$Q_{T,H}\left(A\right)=U_{T,H}\left(\left(\left({(m+{\alpha }_1)}^{-i\tau }:m\in {\mathbb{N}}_0\right),\dots ,\left({(m+{\alpha }_r)}^{-i\tau }:m\in {\mathbb{N}}_0\right)\right)\in A\right).$$

Lemma 1.

Suppose that the set $\mathcal{L}({\alpha }_1,\dots ,{\alpha }_r)$ is linearly independent over $\mathbb{Q}$, and $T^{23/70}⩽H⩽T$. Then we have

$$Q_{T,H}{\displaystyle \underset{T\to \infty }{\overset{\mathrm{w}}{\to }}}\mu .$$

Proof. 

We consider the Fourier transform of the measure $Q_{T,H}$. Recall that the characters $\chi$ of the group ${\Omega }^r$ are given by

$$\chi \left(\omega \right)={\displaystyle \prod _{j=1}^r}\underset{m\in {\mathbb{N}}_0}{{\prod }^{*}}{\omega }_j^{k_{jm}}\left(m\right),$$

where the star * indicates that only a finite number of integer numbers $k_{jm}$ are distinct from zero. Hence, the Fourier transform ${\mathcal{F}}_{T,H}({\underline{k}}_1,\dots ,{\underline{k}}_r)$, ${\underline{k}}_j=(k_{jm}\in \mathbb{Z}:m\in {\mathbb{N}}_0)$, $j=1,\dots ,r$, of the measure $Q_{T,H}$ is

$$\begin{array}{cc}\hfill {\mathcal{F}}_{T,H}({\underline{k}}_1,\dots ,{\underline{k}}_r)& ={\displaystyle \underset{{\Omega }^r}{\int }}\left({\displaystyle \prod _{j=1}^r}{\displaystyle \underset{m\in {\mathbb{N}}_0}{{\prod }^{*}}}{\omega }_j^{k_{jm}}\left(m\right)\right)\mathrm{d}{Q}_{T,H}={\displaystyle \frac{1}{H}}{\displaystyle \underset{T}{\overset{T+H}{\int }}}\left({\displaystyle \prod _{j=1}^r}{\displaystyle \underset{m\in {\mathbb{N}}_0}{{\prod }^{*}}}{(m+{\alpha }_j)}^{-i\tau k_{jm}}\right)\mathrm{d}\tau \hfill \\ & ={\displaystyle \frac{1}{H}}{\displaystyle \underset{T}{\overset{T+H}{\int }}}\exp \left(-i\tau {\displaystyle \sum _{j=1}^r}{\displaystyle \underset{m\in {\mathbb{N}}_0}{{\sum }^{*}}}{k}_{jm}\log (m+{\alpha }_j)\right)\mathrm{d}\tau .\hfill \end{array}$$

(2)

The linear independence of the set $\mathcal{L}({\alpha }_1,\dots ,{\alpha }_r)$ implies that

$$L\stackrel{\mathrm{def}}{=}{\displaystyle \sum _{j=1}^r}\underset{m\in {\mathbb{N}}_0}{{\sum }^{*}}{k}_{jm}\log (m+{\alpha }_j)\ne 0$$

for $({\underline{k}}_1,\dots ,{\underline{k}}_r)\ne (\underline{0},\dots ,\underline{0})$, where $\underline{0}=(0,\dots ,0)$. Therefore, in this case we find by (2)

$${\mathcal{F}}_{T,H}({\underline{k}}_1,\dots ,{\underline{k}}_r)={\displaystyle \frac{\exp (-iLT)-\exp (-iL(T+H\left)\right)}{iHL}}.$$

(3)

Obviously, for $({\underline{k}}_1,\dots ,{\underline{k}}_r)=(\underline{0},\dots ,\underline{0})$,

$${\mathcal{F}}_{T,H}({\underline{k}}_1,\dots ,{\underline{k}}_r)=1.$$

Thus, this and (2), (3) show that

$$\underset{T\to \infty }{\lim }{\mathcal{F}}_{T,H}({\underline{k}}_1,\dots ,{\underline{k}}_r)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}({\underline{k}}_1,\dots ,{\underline{k}}_r)=(\underline{0},\dots ,\underline{0}),\hfill \\ 0\hfill & \mathrm{if}({\underline{k}}_1,\dots ,{\underline{k}}_r)\ne (\underline{0},\dots ,\underline{0}).\hfill \end{array}\right.$$

(4)

Since the group ${\Omega }^r$ is compact, it is the Lévy group. Therefore, weak convergence of probability measures follows from convergence of the corresponding Fourier transforms, see Theorem 1.4.2 of [30]. Thus, $Q_{T,H}$ converges weakly to the measure defined by the Fourier transform (4). Since (4) corresponds the Haar measure, this proves the lemma. □

Now, we introduce the auxiliary Dirichlet series related to Hurwitz zeta-functions. Fix $\kappa >1/2$, and, for $m\in {\mathbb{N}}_0$ and $n\in \mathbb{N}$, set

$$a_n(m,{\alpha }_j)=\exp \left(-{\left(\frac{m+{\alpha }_j}{n}\right)}^{\kappa }\right),j=1,\dots ,r.$$

Thus, $a_n(m,{\alpha }_j)$ exponentially decreese with respect to m. Introduce the Dirichlet series

$${\zeta }_n(s,{\alpha }_j)={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{a_n(m,{\alpha }_j)}{{(m+{\alpha }_j)}^s}},j=1,\dots ,r,$$

which, in view of the above remark on $a_n(m,{\alpha }_j)$, converges absolutely in any half-plane $\sigma >{\sigma }_0$. For brevity, set

$${\underline{\zeta }}_n(s,\underline{\alpha })=\left({\zeta }_n(s,{\alpha }_1),\dots ,{\zeta }_n(s,{\alpha }_j)\right),$$

define

$$P_{T,H,n}\left(A\right)=U_{T,H}\left({\underline{\zeta }}_n(s+i\tau ,\underline{\alpha })\in A\right),A\in \mathcal{B}\left({\mathcal{H}}^r\left(\mathcal{D}\right)\right),$$

and investigate weak convergence of $P_{T,H,n}$ as $T\to \infty$. For this, introduce the mapping $v_n:{\Omega }^r\to {\mathcal{H}}^r\left(\mathcal{D}\right)$ defined by

$$v_n\left(\omega \right)={\underline{\zeta }}_n(s,\underline{\alpha },\omega ),$$

where

$${\underline{\zeta }}_n(s,\underline{\alpha },\omega )=\left({\zeta }_n(s,{\alpha }_1,{\omega }_1),\dots ,{\zeta }_n(s,{\alpha }_r,{\omega }_r)\right)$$

and

$${\zeta }_n(s,{\alpha }_j,{\omega }_j)={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{a_n(m,{\alpha }_j){\omega }_j\left(m\right)}{{(m+{\alpha }_j)}^s}},j=1,\dots ,r.$$

Since the series for ${\zeta }_n(s,{\alpha }_j,{\omega }_j)$, $j=1,\dots ,r$, are absolutely convergent for any $\sigma >{\sigma }_0$, the mapping $v_n$ is continuous; hence, it is $(\mathcal{B}\left({\Omega }^r\right),\mathcal{B}\left({\mathcal{H}}^r\left(\mathcal{D}\right)\right))$-measurable. Therefore, the Haar measure on $({\Omega }^r,\mathcal{B}\left({\Omega }^r\right))$ and mapping $v_n$ define the unique probability measure $\mu v_n^{-1}$ on the space $({\mathcal{H}}^r\left(\mathcal{D}\right),\mathcal{B}\left({\mathcal{H}}^r\left(\mathcal{D}\right)\right))$ given by

$$\mu v_n^{-1}\left(A\right)=\mu \left(v_n^{-1}A\right),A\in \mathcal{B}\left({\mathcal{H}}^r\left(\mathcal{D}\right)\right),$$

where $v_n^{-1}A$ denotes the preimage of a set A.

We use the following assertion on preservation of weak convergence under certain mappings.

Lemma 2.

Suppose that $P_n$, $n\in \mathbb{N}$, and P are probability measures on $(\mathcal{Z},\mathcal{B}(\mathcal{Z}\left)\right)$ and $v:\mathcal{Z}\to \mathbb{Y}$ is a continuous mapping. If $P_n{\displaystyle \underset{n\to \infty }{\overset{\mathrm{w}}{\to }}}P$, then $P_n{v}^{-1}{\displaystyle \underset{n\to \infty }{\overset{\mathrm{w}}{\to }}}P{v}^{-1}$ as well.

The lemma is a partial case of Theorem 5.1 of [31].

We will apply Lemma 2 for the probability measures $P_{T,H,n}$, $Q_{T,H}$ and $\mu$.

Lemma 3.

Suppose that the set $\mathcal{L}({\alpha }_1,\dots ,{\alpha }_r)$ is linearly independent over $\mathbb{Q}$. Then $P_{T,H,n}{\displaystyle \underset{T\to \infty }{\overset{\mathrm{w}}{\to }}}\mu v_n^{-1}$.

Proof. 

By the definitions of $P_{T,H,n}$ and $v_n$, we have

$$\begin{array}{cc}\hfill v_n& \left(\left(\left({(m+{\alpha }_1)}^{-i\tau }:m\in {\mathbb{N}}_0\right),\dots ,\left({(m+{\alpha }_r)}^{-i\tau }:m\in {\mathbb{N}}_0\right)\right)\right)\hfill \\ & =\left({\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{a_n(m,{\alpha }_1){(m+{\alpha }_1)}^{-i\tau }}{{(m+{\alpha }_1)}^s}},\dots ,{\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{a_n(m,{\alpha }_r){(m+{\alpha }_r)}^{-i\tau }}{{(m+{\alpha }_r)}^s}}\right)\hfill \\ & =\left({\zeta }_n(s+i\tau ,{\alpha }_1),\dots ,{\zeta }_n(s+i\tau ,{\alpha }_r)\right)={\underline{\zeta }}_n(s+i\tau ,\underline{\alpha }),\hfill \end{array}$$

and

$$\begin{array}{c}\hfill P_{T,H,n}\left(A\right)=U_{T,H}\left(\left(\left({(m+{\alpha }_1)}^{-i\tau }:m\in {\mathbb{N}}_0\right),\dots ,\left({(m+{\alpha }_r)}^{-i\tau }:m\in {\mathbb{N}}_0\right)\right)\in v_n^{-1}A\right),\\ \hfill A\in \mathcal{B}\left({\mathcal{H}}^r\left(\mathcal{D}\right)\right).\end{array}$$

Thus, we obtain $P_{T,H,n}=Q_{T,H}{v}_n^{-1}$, where $Q_{T,H}$ is from Lemma 1. Therefore, By Lemmas 1 and 2, it follows that $P_{T,H,n}{\displaystyle \underset{T\to \infty }{\overset{\mathrm{w}}{\to }}}\mu v_n^{-1}$. □

Denote $Q_n=\mu v_n^{-1}$. We observe that $Q_n$ is independent of H. $Q_n$ remains the same for every H from the interval $[T^{23/70},T]$.

Our next task is to show that the tuples $\underline{\zeta }(s,\underline{\alpha })$ and ${\underline{\zeta }}_n(s,\underline{\alpha })$ are close in a certain sense. For description of that closeness, a suitable metric in ${\mathcal{H}}^r\left(\mathcal{D}\right)$ is needed. Let, for $f_1,f_2\in \mathcal{H}\left(\mathcal{D}\right)$,

$$d(f_1,f_2)={\displaystyle \sum _{l=1}^{\infty }}{2}^{-l}{\displaystyle \frac{{\sup }_{s\in K_l}|f_1\left(s\right)-f_2\left(s\right)|}{1+{\sup }_{s\in K_l}|f_1\left(s\right)-f_2\left(s\right)|}},$$

where $\{K_l:l\in \mathbb{N}\}\subset \mathcal{D}$ is a sequence of compact embedded subsets of $\mathcal{D}$ such that the strip $\mathcal{D}$ is the union of all $K_l$, and every compact set $K\subset \mathcal{D}$ lies in some $K_l$. Then we have that d is a metric in $\mathcal{H}\left(\mathcal{D}\right)$ inducing its topology of uniform convergence on compacta.

Now let ${\underline{f}}_1=(f_{11},\dots ,f_{1r}),{\underline{f}}_2=(f_{21},\dots ,f_{2r})\in {\mathcal{H}}^r\left(\mathcal{D}\right)$. Then, taking

$$d_r({\underline{f}}_1,{\underline{f}}_2)=\underset{1⩽j⩽r}{\max }d(f_{1j},f_{2j})$$

defines a metric in ${\mathcal{H}}^r\left(\mathcal{D}\right)$ that induces its product topology.

The next lemma shows a closeness of $\underline{\zeta }(s,\underline{\alpha })$ and ${\underline{\zeta }}_n(s,\underline{\alpha })$.

Lemma 4.

Suppose that $T^{23/70}⩽H⩽T$. Then

$$\underset{n\to \infty }{\lim }\underset{T\to \infty }{\mathrm{lim\; sup}}{\displaystyle \frac{1}{H}}{\displaystyle \underset{T}{\overset{T+H}{\int }}}{d}_r\left(\underline{\zeta }(s+i\tau ,\underline{\alpha }),{\underline{\zeta }}_n(s+i\tau ,\underline{\alpha })\right)\mathrm{d}\tau =0.$$

Proof. 

By the definitions of the metrics d and $d_r$, it suffices to prove that, for every compact set $K\subset \mathcal{D}$ and $j=1,\dots ,r$,

$$\underset{n\to \infty }{\lim }\underset{T\to \infty }{\mathrm{lim\; sup}}{\displaystyle \frac{1}{H}}{\displaystyle \underset{T}{\overset{T+H}{\int }}}\underset{s\in K}{\sup }\left|\zeta (s+i\tau ,{\alpha }_j)-{\zeta }_n(s+i\tau ,{\alpha }_j)\right|\mathrm{d}\tau =0,$$

and the latter equality using (1) has been obtained in [26], Lemma 4. □

Let $X_n$, $n\in \mathbb{N}$, and X be ${\mathcal{H}}^r\left(\mathcal{D}\right)$-valued random elements with the corresponding distributions $Q_n$ and Q, respectively. We say that $X_n$ converges to X as $n\to \infty$ in distribution ($X_n{\displaystyle \underset{n\to \infty }{\overset{\mathcal{D}}{\to }}}X$) if $Q_N{\displaystyle \underset{n\to \infty }{\overset{\mathrm{w}}{\to }}}Q$. For convenience, we state a version of Theorem 4.2 of [31] for the space ${\mathcal{H}}^r\left(\mathcal{D}\right)$.

Lemma 5

(See [31]). Suppose that $X_n$, $X_{nl}$ and $Y_n$, $n,l\in \mathbb{N}$ are ${\mathcal{H}}^r\left(\mathcal{D}\right)$-valued random elements on a certain probability space $(\mathbb{T},\mathcal{B}(\mathbb{T}),\nu )$,

$$X_{nl}{\displaystyle \underset{n\to \infty }{\overset{\mathcal{D}}{\to }}}{X}_l,\forall l\in \mathbb{N},X_l{\displaystyle \underset{l\to \infty }{\overset{\mathcal{D}}{\to }}}X,$$

and, for every $\epsilon >0$,

$$\underset{l\to \infty }{\lim }\underset{n\to \infty }{\mathrm{lim\; sup}}\nu \left\{d_r(X_{nl},Y_n)⩾\epsilon \right\}=0.$$

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

3.2. Proof of Theorem 4

Let ${\xi }_{T,H}$ be a random variable defined on the certain probability space $(\mathbb{T},\mathcal{B}(\mathbb{T}),\nu )$ and uniformly distributed in the interval $[T,T+H]$. On $(\mathbb{T},\mathcal{B}(\mathbb{T}),\nu )$, define the ${\mathcal{H}}^r\left(\mathcal{D}\right)$-valued random element

$$X_{T,H,n}=X_{T,H,n}\left(s\right)=\underline{\zeta }(s+i{\xi }_{T,H},\underline{\alpha }),$$

and let $X_n$ be the ${\mathcal{H}}^r\left(\mathcal{D}\right)$-valued random element with the distribution $Q_n=\mu v_n^{-1}$. Then the assertion of Lemma 3 can be rewritten as

$$X_{T,H,n}{\displaystyle \underset{T\to \infty }{\overset{\mathcal{D}}{\to }}}{X}_n.$$

(5)

Now, we discuss the random element $X_n$. In [24], the probability measure

$${\widehat{P}}_T\left(A\right)={\displaystyle \frac{1}{T}}\mathrm{meas}\left\{\tau \in [0,T]:\underline{\zeta }(s+i\tau ,\underline{\alpha })\in A\right\},A\in \mathcal{B}\left({\mathcal{H}}^r\left(\mathcal{D}\right)\right),$$

as $T\to \infty$, was considered, and, in the proof of Theorem 9, it was obtained that there exists a probability measure $\widehat{P}$ on $({\mathcal{H}}^r\left(\mathcal{D}\right),\mathcal{B}\left({\mathcal{H}}^r\left(\mathcal{D}\right)\right))$ such that ${\widehat{P}}_T{\displaystyle \underset{T\to \infty }{\overset{\mathrm{w}}{\to }}}\widehat{P}$. Let

$${\widehat{P}}_{T,n}\left(A\right)={\displaystyle \frac{1}{T}}\mathrm{meas}\left\{\tau \in [0,T]:{\underline{\zeta }}_n(s+i\tau ,\underline{\alpha })\in A\right\},A\in \mathcal{B}\left({\mathcal{H}}^r\left(\mathcal{D}\right)\right).$$

Then, in [24], it was obtained that (Lemma 5 and Theorem 6 of [24])

$${\widehat{P}}_{T,n}{\displaystyle \underset{T\to \infty }{\overset{\mathrm{w}}{\to }}}{Q}_n$$

as well. It is very important that, in the proof of Theorem 9 of [24], it was observed that the measure $\widehat{P}$ coincides with the measure P, given by

$$X_n{\displaystyle \underset{n\to \infty }{\overset{\mathcal{D}}{\to }}}P.$$

(6)

Finally, in the proof of Theorem 4 of [24], P has been identified, using elements of ergodic theory, and it was obtained that $P=P_{\underline{\zeta }}$. Thus,

$$X_n{\displaystyle \underset{n\to \infty }{\overset{\mathcal{D}}{\to }}}{P}_{\underline{\zeta }}.$$

(7)

Define

$$Y_{T,H}=Y_{T,H}\left(s\right)=\underline{\zeta }(s+i{\xi }_{T,H},\underline{\alpha }).$$

It remains to show that

$$Y_{T,H}{\displaystyle \underset{T\to \infty }{\overset{\mathcal{D}}{\to }}}{P}_{\underline{\zeta }}.$$

(8)

For this, we apply Lemma 5. In view of Lemma 4, we obtain that, for any $\epsilon >0$,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{\lim }\underset{T\to \infty }{\mathrm{lim\; sup}}& \nu \left\{d_r(X_{T,H,n},Y_{T,H})⩾\epsilon \right\}\hfill \\ & ⩽\underset{n\to \infty }{\lim }\underset{T\to \infty }{\mathrm{lim\; sup}}{\displaystyle \frac{1}{H\epsilon }}{\displaystyle \underset{T}{\overset{T+H}{\int }}}{d}_r\left(\underline{\zeta }(s+i\tau ,\underline{\alpha }),{\underline{\zeta }}_n(s+i\tau ,\underline{\alpha })\right)\mathrm{d}\tau =0.\hfill \end{array}$$

This, (5) and (7) show that all hypotheses of Lemma 5 are satisfied for the random elements $X_{T,H,n}$, $X_n$ and $Y_{T,H}$. Therefore, we have that the relation (8) is valid. The theorem is proved.

4. Proof of Theorem 1

Theorem 1 is a consequence of Theorem 4. We start with recalling two convenient equivalents of weak convergence. Suppose that $P_n$, $n\in \mathbb{N}$, and P are probability measures on $(\mathcal{Z},\mathcal{B}(\mathcal{Z}\left)\right)$. Denote by $\partial A$ the boundary of $A\in \mathcal{B}\left(\mathcal{Z}\right)$. If $P(\partial A)=0$, then A is called the continuity set of P.

Lemma 6.

The statement $P_n{\displaystyle \underset{n\to \infty }{\overset{\mathrm{w}}{\to }}}P$ is equivalent to every one of the statements:

$1^{\circ }$ Let $A\in \mathcal{B}\left(\mathcal{Z}\right)$ be arbitrary continuity set of P. Then the equality

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

holds;

$2^{\circ }$ Let $G\subset \mathcal{Z}$ be an arbitrary open set. Then the inequality

$$\underset{n\to \infty }{\mathrm{lim\; inf}} P_n\left(G\right)⩾P\left(G\right)$$

holds.

A proof of the lemma can be found in [31], Theorem 2.1.

For convenience, we also remind that a support of the measure P on $(\mathcal{Z},\mathcal{B}(\mathcal{Z}\left)\right)$ is a closed minimal set $S\subset \mathcal{Z}$ such that $P\left(S\right)=1$. The set S is formed of all elements $a\in \mathcal{Z}$ satisfying $P\left(G\right)>0$ for any neighbourhood G of a.

Lemma 7

(See Theorem 11 of [24]). Let the set $\mathcal{L}({\alpha }_1,\dots ,{\alpha }_r)$ be linearly independent over $\mathbb{Q}$. Then the support of the measure $P_{\underline{\zeta }}$ is the space ${\mathcal{H}}^r\left(\mathcal{D}\right)$.

We also recall a version of the Mergelyan theorem as the next lemma.

Lemma 8

(See [32,33]). Let $K\subset \mathbb{C}$ be a compact set having connected complement, and $f\left(s\right)$ a continuous function on K and analytic inside of K. Then, for every $\epsilon >0$, there is a polynomial $p\left(s\right)$ such that

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

Proof of Theorem 1.

Fix $\epsilon >0$. Then, by Lemma 8, there are polynomials $p_1\left(s\right),\dots ,p_r\left(s\right)$ satisfying

$$\underset{1⩽j⩽r}{\sup }\underset{s\in K_j}{\sup }\left|p_j\left(s\right)-f_j\left(s\right)\right|<{\displaystyle \frac{\epsilon }{2}}.$$

(9)

Introduce the set

$${\mathcal{G}}_{\epsilon }=\left\{(g_1,\dots ,g_r)\in {\mathcal{H}}^r\left(\mathcal{D}\right):\underset{1⩽j⩽r}{\sup }\underset{s\in K_j}{\sup }\left|g_j\left(s\right)-p_j\left(s\right)\right|<\frac{\epsilon }{2}\right\}.$$

In virtue of Lemma 7, the set ${\mathcal{G}}_{\epsilon }$ is a neighbourhood of the element $(p_1\left(s\right),\dots ,p_r\left(s\right))$ of the support ${\mathcal{H}}^r\left(\mathcal{D}\right)$ of the measure $P_{\underline{\zeta }}$. Hence, by a support property,

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

(10)

Since ${\mathcal{G}}_{\epsilon }$ is open, by Theorem 4 and $2^{\circ }$ of Lemma 6, we find that

$$\underset{T\to \infty }{\mathrm{lim\; inf}} P_{T,H,\underline{\zeta }}\left({\mathcal{G}}_{\epsilon }\right)⩾P_{\underline{\zeta }}\left({\mathcal{G}}_{\epsilon }\right)>0,$$

and the forms of $P_{T,H.\underline{\zeta }}$ and ${\mathcal{G}}_{\epsilon }$ yield

$$\underset{T\to \infty }{\mathrm{lim\; inf}} U_{T,H}\left(\underset{1⩽j⩽r}{\sup }\underset{s\in K_j}{\sup }\left|\zeta (s+i\tau ,{\alpha }_j)-p_j\left(s\right)\right|<\frac{\epsilon }{2}\right)>0.$$

(11)

Define one more set

$$\widehat{\mathcal{G}}\epsilon =\left\{(g_1,\dots ,g_r)\in {\mathcal{H}}^r\left(\mathcal{D}\right):\underset{1⩽j⩽r}{\sup }\underset{s\in K_j}{\sup }\left|g_j\left(s\right)-f_j\left(s\right)\right|<\epsilon \right\}.$$

Then (9) shows that ${\mathcal{G}}_{\epsilon }\subset {\widehat{\mathcal{G}}}_{\epsilon }$. This remark together with (11) proves the first part of the Theorem 1.

To prove the second part of the theorem, we need to describe the boundary $\partial {\widehat{\mathcal{G}}}_{\epsilon }$ of the set ${\widehat{\mathcal{G}}}_{\epsilon }$. First we observe that the set $\partial {\widehat{\mathcal{G}}}_{\epsilon }$ belongs to the set

$$\begin{array}{cc}\hfill & \left\{(g_1,\dots ,g_r)\in {\mathcal{H}}^r\left(\mathcal{D}\right):\underset{s\in K_1}{\sup }|g_1\left(s\right)-f_1\left(s\right)||=\epsilon ,(g_2,\dots ,g_r)\in {\mathcal{H}}^{r-1}\left(\mathcal{D}\right)\right\}\cup \dots \hfill \\ & \cup \left\{(g_1,\dots ,g_r)\in {\mathcal{H}}^r\left(\mathcal{D}\right):\underset{s\in K_r}{\sup }|g_r\left(s\right)-f_r\left(s\right)||=\epsilon ,(g_1,\dots ,g_{r-1})\in {\mathcal{H}}^{r-1}\left(\mathcal{D}\right)\right\}.\hfill \end{array}$$

Therefore, the boundaries of the set ${\widehat{\mathcal{G}}}_{\epsilon }$ have no common points for different positive $\epsilon$. Hence, by a standard way, we find $P_{\underline{\zeta }}(\partial {\widehat{\mathcal{G}}}_{\epsilon })>0$ for at most a countable set of values of $\epsilon >0$. Thus, the set ${\widehat{\mathcal{G}}}_{\epsilon }$ is a continuity set of $P_{\underline{\zeta }}$ for all but at most countably many $\epsilon >0$. This and application of Theorem 4 and $1^{\circ }$ of Lemma 6 give

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

for all but at most countably many $\epsilon >0$. This, the definitions of $P_{T,H,\underline{\zeta }}$ and ${\widehat{\mathcal{G}}}_{\epsilon }$ together with (10) prove the second statement of the theorem. □

5. Proof of Theorem 2

Lemma 9.

The mapping u is continuous.

Proof. 

Fix a compact set $K\subset \mathcal{D}$. Then in $\mathcal{D}$ there exist an open set $\mathfrak{G}$ and a compact set $\widehat{K}$ such that $K\subset \mathfrak{G}\subset \widehat{K}$. Let $\gamma$ be a simple closed contour lying in $\widehat{K}\setminus \mathfrak{G}$ and enclosing the set K. Moreover, suppose ${\underline{g}}_j=(g_{j1},\dots ,g_{jr})\in {\mathcal{H}}^r\left(\mathcal{D}\right)$, $j=1,2$. Then, by the Cauchy integral formula, for all $s\in K$, we have

$$\begin{array}{cc}\hfill u({\underline{g}}_1\left(s\right))-u({\underline{g}}_2\left(s\right))& ={\displaystyle \sum _{k=1}^r}{c}_k\underset{\gamma }{\int }{\displaystyle \frac{g_{1k}\left(z\right)-g_{2k}\left(z\right)}{{(z-s)}^{m_k+1}}}\mathrm{d}z\ll {\displaystyle \sum _{k=1}^r}|c_k|\underset{\gamma }{\int }{\displaystyle \frac{|g_{1k}\left(z\right)-g_{2k}\left(z\right)|}{{|z-s|}^{m_k+1}}}|\mathrm{d}z|\hfill \\ & {\ll }_{K,\mathfrak{G}}\underset{1⩽k⩽r}{\sup }\underset{s\in K}{\sup }|g_{1k}\left(s\right)-g_{2k}\left(s\right)|.\hfill \end{array}$$

This gives the continuity of u. □

For $A\in \mathcal{B}\left(\mathcal{H}\right(\mathcal{D}\left)\right)$, define

$$R_{T,H,\underline{\zeta }}\left(A\right)=U_{T,H}\left(u\left(\underline{\zeta }(s+i\tau ,\underline{\alpha })\right)\in A\right).$$

Lemma 10.

Suppose that ${\alpha }_j\in (0,1)$, $j=1,\dots ,r$, the set $\mathcal{L}({\alpha }_1,\dots ,{\alpha }_r)$ is linearly independent over $\mathbb{Q}$, and $T^{23/70}⩽H⩽T$. Then $R_{T,H,\underline{\zeta }}{\displaystyle \underset{T\to \infty }{\overset{\mathrm{w}}{\to }}}{P}_{\underline{\zeta }}{u}^{-1}$.

Proof. 

The statement of the lemma is a corollary of Theorem 4, and Lemmas 2 and 9. □

Lemma 11.

Under hypotheses of Lemma 10, the support of $P_{\underline{\zeta }}{u}^{-1}$ is the set $\mathcal{H}\left(\mathcal{D}\right)$.

Proof. 

First of all we have that, for any polynomial $p\left(s\right)$, there is $\underline{g}\in {\mathcal{H}}^r\left(\mathcal{D}\right)$ such that

$$u\left(\underline{g}\right)=p\left(s\right).$$

Hence, we obtain that the preimage $u^{-1}\left(p\right)$ is non-empty for any polynomial p.

Fix $g\in \mathcal{H}\left(\mathcal{D}\right)$ and take its open neighbourhood $\mathcal{G}$. By Lemma 8, there exists a polynomial $p=p\left(s\right)$ such that $p\in \mathcal{G}$. Hence, we have that $u^{-1}\mathcal{G}$ is an open neighbourhood of some $\underline{g}\in {\mathcal{H}}^r\left(\mathcal{D}\right)$. Thus, in view of Lemma 7,

$$P_{\underline{\zeta }}\left(u^{-1}\mathcal{G}\right)>0,$$

so $P_{\underline{\zeta }}{u}^{-1}\left(\mathcal{G}\right)>0$, and this proves the lemma. □

Proof of Theorem 2.

We follow the proof of Theorem 1. For $\epsilon >0$, by Lemma 8, there is a polynomial $p\left(s\right)$ satisfying

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

(12)

Let

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

In virtue of Lemma 11, the set ${\mathcal{G}}_{\epsilon }$ is an open neighbourhood of the element $p\left(s\right)$ of the support of $P_{\underline{\zeta }}{u}^{-1}$. Hence,

$$P_{\underline{\zeta }}{u}^{-1}\left({\mathcal{G}}_{\epsilon }\right)>0.$$

(13)

Moreover, by Lemma 10 and $2^{\circ }$ of Lemma 6, we obtain

$$\underset{T\to \infty }{\mathrm{lim\; inf}} R_{T,H,\underline{\zeta }}\left({\mathcal{G}}_{\epsilon }\right)⩾P_{\underline{\zeta }}{u}^{-1}\left({\mathcal{G}}_{\epsilon }\right)>0,$$

and this implies

$$\underset{T\to \infty }{\mathrm{lim\; inf}} U_{T,H}\left(\underset{s\in K}{\sup }\left|p\left(s\right)-u\left(\underline{\zeta }(s+i\tau ,\underline{\alpha })\right)\right|<\frac{\epsilon }{2}\right)>0.$$

(14)

Introduce one more set:

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

Inequality (12) implies that ${\mathcal{G}}_{\epsilon }\subset {\widehat{\mathcal{G}}}_{\epsilon }$. From this and (14), we have the first assertion of Theorem 2.

In order to prove the second assertion of Theorem 2, we notice that the boundary $\partial {\widehat{\mathcal{G}}}_{\epsilon }$ of ${\widehat{\mathcal{G}}}_{\epsilon }$ is in the set

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

This shows that the sets $\partial {\widehat{\mathcal{G}}}_{{\epsilon }_1}$ and $\partial {\widehat{\mathcal{G}}}_{{\epsilon }_2}$ do not intersect for ${\epsilon }_1\ne {\epsilon }_2$. Hence, $P_{\underline{\zeta }}{u}^{-1}(\partial {\widehat{G}}_{\epsilon })>0$ for at most countably many $\epsilon >0$. In other words, the set ${\widehat{G}}_{\epsilon }$ is a continuity set of the measure $P_{\underline{\zeta }}{u}^{-1}$ for all but at most countably many $\epsilon >0$. Thus, by Lemma 10 and $1^{\circ }$ of Lemma 6, and (13),

$$\underset{T\to \infty }{\lim }{R}_{T,H,\underline{\zeta }}\left({\widehat{\mathcal{G}}}_{\epsilon }\right)=P_{\underline{\zeta }}{u}^{-1}\left({\widehat{\mathcal{G}}}_{\epsilon }\right)⩾P_{\underline{\zeta }}{u}^{-1}\left({\widehat{\mathcal{G}}}_{\epsilon }\right)>0$$

for all but at most countably many $\epsilon >0$, and the definitions of $R_{T,H,\underline{\zeta }}$ and ${\widehat{\mathcal{G}}}_{\epsilon }$ give the second statement of the theorem. □

6. Proof of Theorem 3

The proof of Theorem 3 is based on Theorem 2 and the Rouché theorem; see, for example, [34], which, for the convenience, we formulate below.

Lemma 12.

Suppose that the functions $g_1$ and $g_2$ are analytic in the region G with closed contour $\partial G$. If

$$|g_2\left(s\right)|<|g_1\left(s\right)|,s\in \partial G,$$

then the functions $g_1$ and $g_1+g_2$ have the same number of zeros inside of $\mathcal{G}$, counting multiplicities.

Proof of Theorem 3.

We use the notation

$${\sigma }_0={\displaystyle \frac{{\sigma }_1+{\sigma }_2}{2}}\mathrm{and}{r}_0={\displaystyle \frac{{\sigma }_2-{\sigma }_1}{2}}.$$

We apply the Rouché theorem (Lemma 12) for the functions

$$u\left(\underline{\zeta }(s+i\tau ,\underline{\alpha })\right)-(s-{\sigma }_0)$$

and $(s-{\sigma }_0)$ in the disc $K=|s-{\sigma }_0|⩽r_0$. Take $\epsilon >0$ satisfying

$$\epsilon <{\displaystyle \frac{1}{5}}\underset{|s-{\sigma }_0|=r_0}{\inf }|s-{\sigma }_0|={\displaystyle \frac{r_0}{5}}.$$

(15)

By Theorem 2, there is $\tau$, $T⩽\tau ⩽T+H$, such that

$$\underset{|s-{\sigma }_0|⩽r_0}{\sup }\left|u\left(\underline{\zeta }(s+i\tau ,\underline{\alpha })\right)-(s-{\sigma }_0)\right|<\epsilon .$$

(16)

In view of (15), we see that, for $|s-{\sigma }_0|=r_0$,

$$\left|u\left(\underline{\zeta }(s+i\tau ,\underline{\alpha })\right)-(s-{\sigma }_0)\right|<|s-{\sigma }_0|.$$

Therefore, the functions

$$u\left(\underline{\zeta }(s+i\tau ,\underline{\alpha })\right)-(s-{\sigma }_0)$$

and $(s-{\sigma }_0)$ satisfy the hypotheses of Lemma 12 in the disc K. Obviously, $s-s_0=0$ for the unique $s=s_0\in K$. Hence, the sum

$$\left(u\left(\underline{\zeta }(s+i\tau ,\underline{\alpha })\right)-(s-{\sigma }_0)\right)+s-{\sigma }_0=u\left(\underline{\zeta }(s+i\tau ,\underline{\alpha })\right)$$

possesses the unique zero in K as well. Since, by Theorem 2, the set of $\tau$ satisfying (16) has a positive lower density in $[T,T+H]$, there is $c=c(\underline{\alpha },c_1,c_2)>0$ such that, for sufficiently large T, the function $u\left(\underline{\zeta }(s,\underline{\alpha })\right)$ has more than $cH$ zeros in the region $\{s\in \mathbb{C}:{\sigma }_1<\sigma <{\sigma }_2,T⩽t⩽T+H\}$. The proof is complete. □

7. Conclusions

We obtained in the paper that a tuple of shifts $(\zeta (s+i\tau ,{\alpha }_1),\dots ,\zeta (s+i\tau ,{\alpha }_r))$ of Hurwitz zeta-functions with parameters ${\alpha }_1,\dots ,{\alpha }_r$, such that the set $\{\log (m+{\alpha }_j):m\in {\mathbb{N}}_0,j=1,\dots ,r\}$ is linearly independent over $\mathbb{Q}$ and simultaneously approximating a tuple of analytic functions in $\mathcal{D}=\{s\in \mathbb{C}:1/2<\sigma <1\}$, has a positive lower density in $[T^{23/70},T]$. Moreover, the latter tuple of shifts, with some exception in approximation accuracy, has a positive density. From this, it is derived that shifts of a linear combination of derivatives of the above Hurwitz zeta-functions approximate as well as analytic functions given in $\mathcal{D}$: the set of approximating shifts is also reached in the sense of lower density (and density) in $[T,T+H]$. Moreover, the latter result implies that the mentioned linear combination has more than $cH$, with some $c>0$, zeros lying in the rectangle $\{s\in \mathbb{C}:{\sigma }_1<\sigma <{\sigma }_2,T⩽t⩽T+H\}$ with arbitrary $1/2<{\sigma }_1<{\sigma }_2<1$.

We notice that the results of the paper can be generalized for more general mappings than linear combinations of derivatives. Moreover, we believe that forthcoming papers can extend the results for the so-called periodic zeta-functions $\zeta (s;\mathfrak{a})$ and $\zeta (s,\alpha ;\widehat{\mathfrak{a}})$ given, for $\sigma >1$, by

$$\zeta (s;\mathfrak{a})={\displaystyle \sum _{m=1}^{\infty }}{\displaystyle \frac{a_m}{m^s}}\mathrm{and}\zeta (s,\alpha ;\widehat{\mathfrak{a}})={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{{\widehat{a}}_m}{{(m+\alpha )}^s}},$$

where $\mathfrak{a}=\{a_m:m\in \mathbb{N}\}$ and $\widehat{\mathfrak{a}}=\{{\widehat{a}}_m:m\in {\mathbb{N}}_0\}$ are periodic sequences of complex numbers. This is based on the possibility of obtaining mean square estimates for $\zeta (s;\mathfrak{a})$ and $\zeta (s,\alpha ;\widehat{\mathfrak{a}})$ in short intervals.