Universality in Short Intervals of the Riemann Zeta-Function Twisted by Non-Trivial Zeros

Abstract

Let $0<{\gamma }_1<{\gamma }_2<\cdots ⩽{\gamma }_k⩽\cdots$ be the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function $\zeta (s)$. Using a certain estimate on the pair correlation of the sequence $\{{\gamma }_k\}$ in the intervals $[N,N+M]$ with $N^{1/2+\epsilon }⩽M⩽N$, we prove that the set of shifts $\zeta (s+ih{\gamma }_k)$, $h>0$, approximating any non-vanishing analytic function defined in the strip $\{s\in \mathbb{C}:1/2<\mathrm{Re}s<1\}$ with accuracy $\epsilon >0$ has a positive lower density in $[N,N+M]$ as $N\to \infty$. Moreover, this set has a positive density for all but at most countably $\epsilon >0$. The above approximation property remains valid for certain compositions $F(\zeta (s))$.

1. Introduction

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

$$\zeta (s)=\sum _{m=1}^{\infty }\frac{1}{m^s}=\prod _p{\left(1-\frac{1}{p^s}\right)}^{-1},$$

where the infinite product is taken over all prime numbers, and has analytic continuation over the whole complex plane, except for the point $s=1$ which is a simple pole with residue 1. The function $\zeta (s)$ and its value distribution play an important role not only in analytic number theory but in mathematics in general.

It is well known by a Bohr and Courant work [1] that the set of values of $\zeta (\sigma +it)$ with any fixed $\sigma \in (1/2,1]$ is dense in $\mathbb{C}$. Voronin obtained [2] the infinite-dimensional version of the Bohr–Courant theorem, proving the so-called universality of $\zeta (s)$. This means that every non-vanishing analytic function in the strip $D=\{s\in \mathbb{C}:1/2<\sigma <1\}$ can be approximated by shifts $\zeta (s+i\tau )$. We recall the modern version of the Voronin theorem. Denote by $\mathcal{K}$ the class of compact subsets of the strip D with connected complements, and by $H_0(K)$ with $K\in \mathcal{K}$ the class of continuous non-vanishing functions on K that are analytic in the interior of K. Then, for $K\in \mathcal{K}$, $f(s)\in H_0(K)$ and every $\epsilon >0$, the inequality

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

is true; see, for example, [3,4,5,6]. Thus, we have that there are infinitely many shifts $\zeta (s+i\tau )$ approximating a given function $f(s)\in H_0(K)$.

The above theorem is of continuous type because $\tau$ in shifts $\zeta (s+i\tau )$ can take arbitrary real values. If $\tau$ runs over a certain discrete set, then we have the discrete universality that was proposed in [7]. Denote by $\#A$ the cardinality of a set A, and suppose that N runs over the set of non-negative integers. If K and $f(s)$ are as above, then we have, for $h>0$ and $\epsilon >0$,

$$\underset{N\to \infty }{lim\; inf}\frac{1}{N+1}\#\left\{0⩽k⩽N:\underset{s\in K}{sup}|\zeta (s+ikh)-f(s)|<\epsilon \right\}>0.$$

Approximations of analytic functions by more general discrete shifts were considered in [8,9,10].

Denote by ${\gamma }_1<{\gamma }_2<\cdots ⩽{\gamma }_k⩽\cdots$ the positive imaginary parts of non-trivial zeros ${\rho }_k={\beta }_k+i{\gamma }_k$ of the function $\zeta (s)$. Discrete universality theorems with shifts $\zeta (s+ih{\gamma }_k)$ were obtained in [11,12]. In [11], for this the Riemann hypothesis was used, while in [12], the weak form of the Montgomery pair correlation conjecture [13] was involved. More precisely, the estimate, for $c>0$,

$$\underset{|{\gamma }_k-{\gamma }_l|<c/logT}{\sum _{0<{\gamma }_k,{\gamma }_l⩽T}}1{\ll }_{c}TlogT,T\to \infty ,$$

(1)

was required. Analogical results for more general functions were given in [14,15].

On the other hand, all above theorems are non-effective in the sense that any concrete shift approximating a given analytic function is not known. This shortcoming leads to the idea of universality in intervals as short as possible containing $\tau$ with approximating property. The first result in this direction was obtained in [16].

Theorem 1.

Suppose that $T^{1/3}{(logT)}^{26/15}⩽H⩽T$, $K\in \mathcal{K}$ and $f(s)\in H_0(K)$. Then, for every $\epsilon >0$,

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

The aim of this paper is the universality of the function $\zeta (s)$ in short intervals with shifts $\zeta (s+ih{\gamma }_k)$. In this case, the estimate (1) is not sufficient. Therefore, for $N^{1/2+\epsilon }⩽M⩽N$ with $\epsilon >0$, we use the following hypothesis:

$$\underset{|{\gamma }_k-{\gamma }_l|<c/logN}{\sum _{k=N}^{N+M}\sum _{l=N}^{N+M}}1{\ll }_{c}M,$$

(2)

which, as estimate (1), also gives a certain information on the pair correlation of non-trivial zeros, differently from estimate (1), however, in short intervals.

Theorem 2.

Suppose that $N^{1/2+\epsilon }⩽M⩽N$, and estimate (2) are true. Let $K\in \mathcal{K}$ and $f(s)\in H_0(K)$. Then, for every $\epsilon >0$ and $h>0$,

$$\underset{N\to \infty }{lim\; inf}\frac{1}{M+1}\#\left\{N⩽k⩽N+M:\underset{s\in K}{sup}|\zeta (s+ih{\gamma }_k)-f(s)|<\epsilon \right\}>0.$$

Moreover, “lim inf” can be replaced by “lim” for all but at most countably many $\epsilon >0$.

Theorem 2 has a generalization for certain compositions $F(\zeta (s))$. Denote by $H(D)$ the space of analytic functions on the strip D endowed with the topology of uniform convergence on compacta. Moreover, let

$$S=\left\{g\in H(D):\mathrm{either}g(s)\ne 0\mathrm{for}\mathrm{all}s\in D,\mathrm{or}g(s)\equiv 0\right\},$$

and, for the operator $F:H(D)\to H(D)$ and distinct complex numbers $a_1,\dots ,a_r$,

$$H_{a_1,\dots ,a_r;F}(D)=\left\{g\in H(D):g(s)\ne a_j\mathrm{for}\mathrm{all}s\in D,j=1,\dots ,r\right\}\cup \{F(0)\}.$$

Then we have

Theorem 3.

Suppose that estimate (2) is true, $N^{1/2+\epsilon }⩽M⩽N$, and $F:H(D)\to H(D)$ is a continuous operator such that $F(S)\supset H_{a_1,\dots ,a_r;F}(D)$. For $r=1$, let $K\in \mathcal{K}$ and $f(s)$ be a continuous $\ne a_1$ function on K, and analytic in the interior of K. For $r⩾2$, let K be an arbitrary compact subset of D, and $f(s)\in H_{a_1,\dots ,a_r;F}(D)$. Then, for every $\epsilon >0$ and $h>0$,

$$\underset{N\to \infty }{lim\; inf}\frac{1}{M+1}\#\left\{N⩽k⩽N+M:\underset{s\in K}{sup}|F(\zeta (s+ih{\gamma }_k))-f(s)|<\epsilon \right\}>0.$$

Moreover “lim inf” can be replaced by “lim” for all but at most countably many $\epsilon >0$.

For example, the operators $F(g)=sing$ and $F(g)=sinhg$ satisfy the hypotheses of Theorem 3 with $a_1=-1$ and $a_2=1$.

The proofs of Theorems 2 and 3 use probabilistic limit theorems for measures in the space $H(D)$. Denote by $\mathcal{B}(\mathbb{X})$ the Borel $\sigma$-field of the space $\mathbb{X}$. The main limit theorem will be proved for

$$P_{N,M,h}(A)=\frac{1}{M+1}\#\left\{N⩽k⩽N+M:\zeta (s+ih{\gamma }_k)\in A\right\},A\in \mathcal{B}(H(D)),$$

as $N\to \infty$. We divide its proof into four sections.

2. A Limit Theorem on the Torus

Denote by $\gamma$ the unit circle on the complex plane, by $\mathbb{P}$ the set of all prime numbers, and define the set

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

where ${\gamma }_p=\gamma$ for all $p\in \mathbb{P}$. With the product topology and pointwise multiplication, the torus $\Omega$ is a compact topological Abelian group. Therefore, on $(\Omega ,\mathcal{B}(\Omega ))$, the probability Haar measure $m_H$ can be defined, and we have the probability space $(\Omega ,\mathcal{B}(\Omega ),m_H)$. Denote by $\omega (p)$ the pth component of an element $\omega \in \Omega$, $p\in \mathbb{P}$.

In this section, we will prove a limit theorem for

$$Q_{N,M,h}(A)=\frac{1}{M+1}\#\left\{N⩽k⩽N+M:\left(p^{-ih{\gamma }_k}:p\in \mathbb{P}\right)\in A\right\},A\in \mathcal{B}(\Omega ),$$

as $N\to \infty$.

Before the statement of a limit theorem for $Q_{N,M,h}$ as $N\to \infty$, we will recall some useful results that will be used in its proof. Denote by $N(T)$ the number of non-trivial zeros of $\zeta (s)$ in the region $\{s\in \mathbb{C}:0<t<T\}$.

Lemma 1 (von Mongoldt formula).

For $T\to \infty$,

$$N(T)=\frac{T}{2\pi }log\frac{T}{2\pi \mathrm{e}}+O(logT).$$

For the proof, see, for example, [17].

Denote by $N(\sigma ,T)$ the number of zeros $\mathit{\rho }=\mathit{\beta }+\mathit{i}\mathit{\gamma }$ of $\zeta (s)$ with $\beta >\sigma$ and $|\gamma |<T$.

Lemma 2.

Suppose that $H⩾T^{\alpha }$ with $\alpha >27/82$. Then, for $1/2<\sigma <1$, uniformly in σ,

$$N(\sigma ,T+H)-N(\sigma ,T)=O\left(\frac{H}{\sigma -1/2}\right).$$

Proof of the lemma can be found in [18].

For positive $u\ne 1$, denote by $\Lambda (u)$ the von Mongoldt function if $u\in \mathbb{N}\setminus \{1\}$, and zero, otherwise.

Lemma 3.

For positive $x\ne 1$ and $T\to \infty$,

$$\sum _{0<{\gamma }_k<T}{x}^{{\rho }_k}=\left(\Lambda (x)-x\Lambda \left(\frac{1}{x}\right)\right)\frac{T}{2\pi }+O\left(T^{(1/2)+\epsilon }\right)$$

with every $\epsilon >0$.

Proof. 

The lemma is Theorem 2 of [19] with $a=0$. □

Lemma 4.

Suppose that $N^{1/2+\epsilon }⩽M⩽N$ with $\epsilon >0$. Then, for positive $x\ne 1$, as $N\to \infty$,

$$\sum _{k=N}^{N+M}{x}^{{\rho }_k}{\ll }_x\frac{M}{\sqrt{logM}}.$$

Proof. 

Since

$$\frac{N}{logN}\ll {\gamma }_N\ll \frac{N}{logN},$$

in view of Lemma 3,

$$\sum _{{\gamma }_N<\gamma ⩽{\gamma }_{N+M}}{x}^{\rho }=\left(\Lambda (x)-x\Lambda \left(\frac{1}{x}\right)\right)\frac{{\gamma }_{N+M}-{\gamma }_N}{2\pi }+O\left(\frac{N^{1/2+\epsilon }}{\sqrt{logN}}\right).$$

(3)

An application of Lemma 1 gives

$$N+M=\sum _{\gamma ⩽{\gamma }_{N+M}}1=\frac{{\gamma }_{N+M}}{2\pi }log\frac{{\gamma }_{N+M}}{2\pi \mathrm{e}}+O(logN)$$

and

$$N=\sum _{\gamma ⩽{\gamma }_N}1=\frac{{\gamma }_N}{2\pi }log\frac{{\gamma }_N}{2\pi \mathrm{e}}+O(logN).$$

Therefore,

$${\gamma }_{N+M}=\frac{2\pi (N+M)}{log({\gamma }_{N+M}/(2\pi \mathrm{e}))}+O(1)$$

and

$${\gamma }_N=\frac{2\pi N}{log({\gamma }_N/(2\pi \mathrm{e}))}+O(1).$$

Hence,

$${\gamma }_{N+M}-{\gamma }_N⩽\frac{2\pi (N+M)}{log({\gamma }_N/(2\pi \mathrm{e}))}-\frac{2\pi N}{log({\gamma }_N/(2\pi \mathrm{e}))}+O(1)\ll \frac{M}{logN}+O(1)\ll \frac{M}{logM}.$$

(4)

This together with Equation (3) proves the lemma. □

Now, we state the limit theorem for $Q_{N,M,h}$.

Theorem 4.

Suppose that, for any $\epsilon >0$, $N^{1/2+\epsilon }⩽M⩽N$. Then $Q_{N,M,h}$ converges weakly to the Haar measure $m_H$ as $N\to \infty$.

Proof. 

Denote by $g_{N,M,h}(\underline{k})$, $\underline{k}=(k_p:k_p\in \mathbb{Z},p\in \mathbb{P})$, the Fourier transform of $Q_{N,M,h}$, i.e.,

$$g_{N,M,h}(\underline{k})={\int }_{\Omega }\left(\underset{p\in \mathbb{P}}{{\prod }^{*}}{\omega }^{k_p}(p)\right)\mathrm{d}{Q}_{N,M,h},$$

where the star “*” means that only a finite number of integers $k_p$ are distinct from zero. Thus, by the definition of $Q_{N,M,h}$,

$$g_{N,M,h}(\underline{k})=\frac{1}{M+1}\sum _{k=N}^{N+M}exp\left\{-ih{\gamma }_k\underset{p\in \mathbb{P}}{{\sum }^{*}}{k}_{p}logp\right\}.$$

(5)

Clearly,

$$g_{N,M,h}(\underline{0})=1.$$

(6)

Now, suppose that $k\ne \underline{0}$. Since the set $\{logp:p\in \mathbb{P}\}$ is linearly independent within the field of rational numbers $\mathbb{Q}$, in that case we have

$$a\stackrel{def}{=}\underset{p\in \mathbb{P}}{{\sum }^{*}}{k}_{p}logp\ne 0.$$

Thus, we will estimate the sum

$$\sum _{k=N}^{N+M}exp\{iha{\gamma }_k\}.$$

It is easily seen that

$$\begin{array}{cc}\hfill \sum _{k=N}^{N+M}\left(exp\{ha{\beta }_k\}-exp\left\{\frac{1}{2}ha\right\}\right)& {\ll }_{h,a}\sum _{k=N}^{N+M}\left|exp\left\{ha\left({\beta }_k-\frac{1}{2}\right)\right\}-1\right|\hfill \\ & {\ll }_{h,a}\sum _{k=N}^{N+M}\left|{\beta }_k-\frac{1}{2}\right|=\underset{k=N}{\overset{N+M}{{\sum }^{\prime }}}\left|{\beta }_k-\frac{1}{2}\right|+\underset{k=N}{\overset{N+M}{{\sum }^{″}}}\left|{\beta }_k-\frac{1}{2}\right|,\hfill \end{array}$$

(7)

where $|{\beta }_k-1/2|⩽1/loglogM$ in ${\sum }^{\prime }$, and $|{\beta }_k-1/2|>1/loglogM$ in ${\sum }^{″}$. Obviously,

$$\underset{k=N}{\overset{N+M}{{\sum }^{\prime }}}\left|{\beta }_k-\frac{1}{2}\right|⩽\frac{M}{loglogM}.$$

(8)

Therefore, by Lemma 2 and estimate (2),

$$\underset{k=N}{\overset{N+M}{{\sum }^{″}}}\left|{\beta }_k-\frac{1}{2}\right|\ll \underset{{\gamma }_N<\gamma ⩽{\gamma }_{N+M}}{{\sum }^{″}}1\ll \frac{MloglogM}{logM}.$$

This, and estimates (7) and (8) show that

$$\sum _{k=N}^{N+M}exp\{({\beta }_k+i{\gamma }_k)ha\}-\sum _{k=N}^{N+M}exp\left\{\left(\frac{1}{2}+i{\gamma }_k\right)ha\right\}{\ll }_{h,a}\frac{M}{loglogM}.$$

(9)

Lemma 4 with $x=exp\{ha\}$ implies

$$\sum _{k=N}^{N+M}exp\{({\beta }_k+i{\gamma }_k)ha\}{\ll }_{h,a}\frac{M}{\sqrt{logM}}.$$

Therefore, in view of estimate (9),

$$\sum _{k=N}^{N+M}exp\{iha{\gamma }_k\}{\ll }_{h,a}\sum _{k=N}^{N+M}exp\left\{\left(\frac{1}{2}+i{\gamma }_k\right)ha\right\}{\ll }_{h,a}\frac{M}{loglogM}.$$

Thus, by Equation (5),

$$g_{N,M,h}(\underline{k}){\ll }_{h,a}\frac{1}{loglogM}.$$

This together with Equation (6) shows that

$$\underset{N\to \infty }{lim}{g}_{N,M,h}(\underline{k})=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\underline{k}=\underline{0},\hfill \\ 0\hfill & \mathrm{if}\underline{k}\ne \underline{0},\hfill \end{array}\right.$$

and the lemma is proved because the right-hand side of the latter equality is the Fourier transform of the measure $m_H$. □

3. A Limit Theorem for Absolutely Convergent Series

Let $\theta >1/2$ be a fixed number, and $v_n(m)=exp\{-{(m/n)}^{\theta }\}$ for $m,n\in \mathbb{N}$. Extend the function $\omega (p)$ to the set $\mathbb{N}$ by setting

$$\omega (m)=\underset{p^{l+1}\nmid m}{\prod _{p^l\mid m}}{\omega }^l(p),$$

and define

$${\zeta }_n(s)=\sum _{m=1}^{\infty }\frac{v_n(m)}{m^s}$$

and

$${\zeta }_n(s,\omega )=\sum _{m=1}^{\infty }\frac{\omega (m)v_n(m)}{m^s}.$$

Then the latter series are absolutely convergent for $\sigma >1/2$ [5]. Consider the function $u_n:\Omega \to H(D)$ defined by

$$u_n(\omega )={\zeta }_n(s,\omega ).$$

The absolute convergence of the series ${\zeta }_n(s,\omega )$ implies the continuity of $u_n$.

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

$$P_{N,M,n,h}(A)=\frac{1}{M+1}\#\left\{N⩽k⩽N+M:{\zeta }_n(s+ih{\gamma }_k)\in A\right\}.$$

Theorem 5.

Suppose that $N^{1/2+\epsilon }⩽M⩽N$. Then $P_{N,M,n,h}$ converges weakly to the measure $m_H{u}_n^{-1}\stackrel{def}{=}{V}_n$.

Proof. 

The theorem follows from the equality

$$P_{N,M,n,h}(A)=Q_{N,M,h}(u_n^{-1}A)=Q_{N,M,h}{u}_n^{-1}(A),A\in \mathcal{B}(H(D)),$$

continuity of the function $u_n$, Theorem 4 and Theorem 5.1 of [20]. □

The weak convergence of $P_{N,M,h}$ is closely connected to that of $V_n$ as $n\to \infty$. Define

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

Then $\zeta (s,\omega )$ is an $H(D)$-valued random element on the probability space $(\Omega ,\mathcal{B}(\Omega ),m_H)$ [5]. We recall that the latter infinite product, for almost all $\omega$, is uniformly convergent on compact subsets $K\subset D$. Denote by $P_{\zeta }$ the distribution of the random element $\zeta (s,\omega )$, i.e.,

$$P_{\zeta }(A)=m_H\{\omega \in \Omega :\zeta (s,\omega )\in A\},A\in \mathcal{B}(H(D)).$$

The following statement is very important.

Proposition 1.

The probability measure $V_n$ converges weakly to measure $P_{\zeta }$ as $n\to \infty$.

Proof. 

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

$$R_T(A)=\frac{1}{T}\mathrm{meas}\{\tau \in [0,T]:\zeta (s+i\tau )\in A\}.$$

It is known that $R_T$, as $T\to \infty$, converges weakly to $P_{\zeta }$ [5]. Moreover, $R_T$, as $T\to \infty$, and $V_n$, as $n\to \infty$, converge weakly to the same probability measure on $(H(D),\mathcal{B}(H(D)))$. Thus, $V_n$ converges weakly to $P_{\zeta }$ as $n\to \infty$. □

4. Mean Square Estimates in Short Intervals

To derive the weak convergence of $P_{N,M,h}$ from that of $P_{N,M,n,h}$ as $N\to \infty$, the estimate for

$$\sum _{k=N}^{N+M}{|\zeta (\sigma +ih{\gamma }_k+it)|}^2$$

with $t\in \mathbb{R}$ is needed.

We will use the following mean square estimate in short intervals.

Lemma 5.

Suppose that σ, $1/2<\sigma <1$, is fixed and $T^{1/3}{(logT)}^{26/15}⩽H⩽T$. Then, uniformly in H,

$${\int }_{T-H}^{T+H}{|\zeta (\sigma +it)|}^2{\ll }_{\sigma }H.$$

The lemma follows from Theorem 7.1 of [21], and was used in [16].

Lemma 6.

Suppose that $N^{1/2+\epsilon }⩽M⩽N$ and estimate (2) is true. Then, for every fixed σ, $1/2<\sigma <1$, $h>0$ and $t\in \mathbb{R}$,

$$\sum _{k=N}^{N+M}|\zeta (\sigma +ih{\gamma }_k+it)|{\ll }_{\sigma ,h}M(1+|t|).$$

Proof. 

We will apply the Gallagher lemma connecting discrete mean squares with those continuous of some functions; for the proof, see Lemma 1.4 of [22]. Let $T_0$, $T⩾\delta >0$ be real numbers, $\mathcal{T}\ne Ø$ be a finite set in the interval $\left[T_0+\delta /2,T_0+T-\delta /2\right]$,

$$N_{\delta }(x)=\underset{|t-x|<\delta }{\sum _{t\in \mathcal{T}}}1$$

and let $S(x)$ be a complex-valued continuous function on $[T_0,T+T_0]$ having a continuous derivative on $(T_0,T+T_0)$. Then the Gallagher lemma asserts that

$$\sum _{t\in \mathcal{T}}{N}_{\delta }^{-1}{(t)|S(t)|}^2⩽\frac{1}{\delta }\underset{T_0}{\overset{T_0+T}{\int }}{|S(x)|}^2\mathrm{d}x+{\left(\underset{T_0}{\overset{T_0+T}{\int }}{|S(x)|}^2\mathrm{d}x\underset{T_0}{\overset{T_0+T}{\int }}{|S^{\prime }(x)|}^2\mathrm{d}x\right)}^{1/2}.$$

(10)

We apply the Gallagher lemma for the function $\zeta (s+ikh{\gamma }_k+it)$. In our case $\delta =c/logN$, $T_0=h{\gamma }_N-\delta /2$, $T=h{\gamma }_{N+M}-h{\gamma }_N+\delta /2$ and $\mathcal{T}=\{h{\gamma }_N,h{\gamma }_{N+1},\dots ,$$h{\gamma }_{N+M}$}. By estimate (2), we have

$$\sum _{k=N}^{N+M}{N}_{\delta }(h{\gamma }_k)=\underset{|{\gamma }_k-{\gamma }_l|<c/(hlogN)}{\sum _{k=N}^{N+M}\sum _{l=N}^{N+M}}1{\ll }_{h}M.$$

(11)

Now, an application of the Gallagher lemma gives

$$\begin{array}{cc}\hfill \sum _{k=N}^{N+M}|\zeta (\sigma +ih{\gamma }_k+it)|& =\sum _{k=N}^{N+M}\sqrt{N_{\delta }(h{\gamma }_k)N^{-1}(h{\gamma }_k)}|\zeta (\sigma +ih{\gamma }_k+it)|\hfill \\ & ⩽{\left(\sum _{k=N}^{N+M}{N}_{\delta }(h{\gamma }_k)\sum _{k=N}^{N+M}{N}^{-1}(h{\gamma }_k){|\zeta (\sigma +ih{\gamma }_k+it)|}^2\right)}^{1/2}\hfill \\ & {\ll }_h\sqrt{M}\sqrt{logN}\left({\int }_{h{\gamma }_N-\delta /2}^{h{\gamma }_{N+M}}{|\zeta (\sigma +i\tau +it)|}^2\mathrm{d}\tau \right.\hfill \\ & +{\left.{\left({\int }_{h{\gamma }_N-\delta }^{h{\gamma }_{N+M}}{|\zeta (\sigma +i\tau +it)|}^2\mathrm{d}\tau {\int }_{h{\gamma }_N-\delta }^{h{\gamma }_{N+M}}{|{\zeta }^{\prime }(\sigma +i\tau +it)|}^2\mathrm{d}\tau \right)}^{1/2}\right)}^{1/2}.\hfill \end{array}$$

(12)

The estimate (4) gives with certain $c_h>0$

$${\int }_{h{\gamma }_N-\delta }^{h{\gamma }_{N+M}}{|\zeta (\sigma +i\tau +it)|}^2\mathrm{d}t\ll {\int }_{h{\gamma }_N-\delta -|t|}^{h{\gamma }_N+c_h(M/logM)+|t|}{|\zeta (\sigma +i\tau )|}^2\mathrm{d}\tau .$$

(13)

If $c_h(M/logM)+|t|⩽h{\gamma }_N$, then, in view of Lemma 5, the right-hand side of (13) is

$${\ll }_{\sigma ,h}\frac{M}{logM}+|t|{\ll }_{\sigma ,h}\frac{M}{logM}(1+|t|).$$

If $c_h(M/logM)+|t|>h{\gamma }_N$, then

$$h{\gamma }_N+c_h\frac{M}{logM}+|t|<2\left(c_h\frac{M}{logM}+|t|\right)$$

and

$$h{\gamma }_N-\delta >h{\gamma }_N-2c_h\frac{M}{logM}-2|t|>-h{\gamma }_N{c}_h\frac{M}{logM}-|t|>-2\left(c_h\frac{M}{logM}+|t|\right).$$

Thus, in this case,

$${\int }_{h{\gamma }_N-\delta }^{h{\gamma }_{N+M}}{|\zeta (\sigma +i\tau +it)|}^2\mathrm{d}\tau {\ll }_h{\int }_{-2(c_h(M/logM)+|t|)}^{2(c_h(M/logM)+|t|)}{|\zeta (\sigma +i\tau )|}^2\mathrm{d}\tau {\ll }_{\sigma ,h}\frac{M}{logM}(1+|t|).$$

This together with estimate (13) shows that

$${\int }_{h{\gamma }_N-\delta }^{h{\gamma }_{N+M}}{|\zeta (\sigma +i\tau +it)|}^2\mathrm{d}\tau {\ll }_{\sigma ,h}\frac{M}{logM}(1+|t|).$$

(14)

Estimate (14) and an application of the Cauchy integral formula lead to the bound

$${\int }_{h{\gamma }_N-\delta }^{h{\gamma }_{N+M}}|{\zeta }^{\prime }{(\sigma +i\tau +it)|}^2\mathrm{d}\tau {\ll }_{\sigma ,h}\frac{M}{logM}(1+|t|).$$

This, estimate (14) and (12) prove the lemma. □

Now, we are ready to state an approximation lemma.

5. Approximation in the Mean

Denote by $\rho$ the metric in $H(D)$ which induces the topology of uniform convergence on compacta. More precisely, for $g_1,g_2\in H(D)$,

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

where $\{K_l:l\in \mathbb{N}\}\subset D$ is a sequence of compact subsets such that

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

$K_l\subset K_{l+1}$ for all $l\in \mathbb{N}$, and every compact $K\subset D$ lies in a certain $K_l$.

Lemma 7.

Suppose that $N^{1/2+\epsilon }⩽M⩽N$ and (2) is true. Then, for every $h>0$,

$$\underset{n\to \infty }{lim}\underset{N\to \infty }{lim\; sup}\frac{1}{M+1}\sum _{k=N}^{N+M}\rho \left(\zeta (s+ih{\gamma }_k),{\zeta }_n(s+ih{\gamma }_k)\right)=0.$$

Proof. 

In view of the definition of the metric $\rho$, it suffices to show that, for every compact $K\subset D$,

$$\underset{n\to \infty }{lim}\underset{N\to \infty }{lim\; sup}\frac{1}{M+1}\sum _{k=N}^{N+M}\underset{s\in K}{sup}\left|\zeta (s+ih{\gamma }_k)-{\zeta }_n(s+ih{\gamma }_k)\right|=0.$$

(15)

Thus, let $K\subset D$ be a fixed compact set. Denote the points of K by $s=\sigma +iv$, and fix $\epsilon >0$ such that $\mathbf{1}/\mathbf{2}+\mathbf{2}\mathit{\epsilon }⩽\mathit{\sigma }⩽\mathbf{1}-\mathit{\epsilon }$ for $s\in K$. It is known [5] that

$${\zeta }_n(s)=\frac{1}{2\pi i}{\int }_{\theta -i\infty }^{\theta +i\infty }\zeta (s+z)l_n(z)\frac{\mathrm{d}z}{z},$$

where

$$l_n(s)=\frac{s}{\theta }\Gamma (s/\theta )n^s,$$

$\Gamma (s)$ is the Euler gamma-function, and $\theta$ comes from the definition of $v_n(m)$. Let ${\theta }_1>0$. From this, we have

$$\zeta (s)-{\zeta }_n(s)=\frac{1}{2\pi i}{\int }_{-\theta -i\infty }^{-\theta +i\infty }\zeta (s+z)l_n(z)\frac{\mathrm{d}z}{z}+R_n(s),$$

with

$$R_n(s)=\frac{l_n(1-s)}{1-s}.$$

Therefore, as in the proof of Lemma 12 of [16], we find that

$$\begin{array}{cc}\hfill \frac{1}{M+1}& \sum _{k=N}^{N+M}\underset{s\in K}{sup}\left|\zeta (s+ih{\gamma }_k)-{\zeta }_n(s+ih{\gamma }_k)\right|\hfill \\ & \ll {\int }_{-\infty }^{\infty }\frac{1}{M}\sum _{k=N}^{N+M}\left|\zeta \left(\frac{1}{2}+\epsilon +i(h{\gamma }_k+t)\right)\right|\underset{s\in K}{sup}\frac{|l_n(1/2+\epsilon -s+it)|}{|1/2+\epsilon -s+it|}\mathrm{d}t\hfill \\ & +\frac{1}{M}\sum _{k=N}^{N+M}\underset{s\in K}{sup}|R_n(s+ih{\gamma }_k)|\stackrel{def}{=}{I}_1+I_2.\hfill \end{array}$$

(16)

Denote by $c_1,c_2,\dots$ positive constants. In view of the well-known estimate

$$\Gamma (\sigma +it)\ll exp\{-c_1|t|\},$$

(17)

we find that

$$\frac{|l_n(1/2+\epsilon -s+it)|}{|1/2+\epsilon -s+it|}\ll n^{-\epsilon }exp\{-c_2|t-v|\}{\ll }_{K,\epsilon }{n}^{-\epsilon }exp\{-c_3|t|\}.$$

Therefore, by Lemma 5,

$$I_1{\ll }_{K,\epsilon }{n}^{-\epsilon }{\int }_{-\infty }^{\infty }(1+|t|)exp\{-c_3|t|\}\mathrm{d}t{\ll }_{K,\epsilon }{n}^{-\epsilon }.$$

(18)

Similarly, taking into account inequality (17), we find

$$\begin{array}{cc}\hfill I_2& \ll \frac{n^{1/2-2\epsilon }}{M}\sum _{k=N}^{N+M}exp\{-c_4|h{\gamma }_k-v|\}{\ll }_K\frac{n^{1/2-2\epsilon }}{M}\sum _{k=N}^{N+M}exp\{-c_{5}h{\gamma }_k\}\hfill \\ & {\ll }_K\frac{n^{1/2-2\epsilon }}{M}\sum _{k=N}^{N+M}exp\{-c_{6}h(k/logk)\}{\ll }_{K,h}\frac{n^{1/2-2\epsilon }}{M}.\hfill \end{array}$$

This, Equations (18) and (16) prove (15). □

6. A Limit Theorem for $\mathit{\zeta }\mathbf{(}\mathit{s}\mathbf{)}$

Using the results of Section 3 and Section 4 leads to a limit theorem for $P_{N,M,h}$.

Theorem 6.

Suppose that $N^{1/2+\epsilon }⩽M⩽N$ and estimate (2) is true. Then $P_{N,M,h}$ converges weakly to $P_{\zeta }$ as $N\to \infty$.

Proof. 

In a certain probability space with measure $\mu$ define the random variable ${\theta }_{N,M,h}$ with the distribution

$$\mu \{{\theta }_{N,M,h}=h{\gamma }_k\}=\frac{1}{M+1},k=N,N+1,\dots ,N+M,$$

and consider the $H(D)$-valued random element

$$X_{N,M,n,h}=X_{N,M,n,h}(s)={\zeta }_n(s+i{\theta }_{N,M,h}).$$

Moreover, let $X_n=X_n(s)$ be the $H(D)$-valued random element with the distribution $V_n$. Then, by Theorem 5,

$$X_{N,M,n,h}\underset{N\to \infty }{\overset{\mathcal{D}}{\to }}{X}_n,$$

(19)

where $\underset{}{\overset{\mathcal{D}}{\to }}$ denotes the convergence in distribution. Moreover, by Proposition 1,

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

(20)

Define one more $H(D)$-valued random element

$$X_{N,M,h}=X_{N,M,h}(s)=\zeta (s+i{\theta }_{N,M,h}).$$

Then, using Lemma 7, we find that, for every $\epsilon >0$,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}& \underset{N\to \infty }{lim\; sup}\mu \left\{\rho (X_{N,M,h},X_{N,M,n,h})⩾\epsilon \right\}\hfill \\ & ⩽\underset{n\to \infty }{lim}\underset{N\to \infty }{lim\; sup}\frac{1}{\epsilon (M+1)}\sum _{k=N}^{N+M}\rho (\zeta (s+ih{\gamma }_k),{\zeta }_n(s+ih{\gamma }_k))=0.\hfill \end{array}$$

Now, this, Equations (19) and (20) together with Theorem 4.2 of [20] show that

$$X_{N,M,h}\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}{P}_{\zeta },$$

and theorem is proved. □

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

$$P_{N,M,h,F}(A)=\frac{1}{M+1}\#\left\{N⩽k⩽N+M:F(\zeta (s+ih{\gamma }_k))\in A\right\}.$$

Corollary 1.

Suppose that $F:H(D)\to H(D)$ is a continuous operator, and (2) is true. Then $P_{N,M,h,F}$ converges weakly to $P_{\zeta }{F}^{-1}$ as $N\to \infty$.

Proof. 

The corollary follows from Theorem 5, continuity of F, equality

$$P_{N,M,h,F}=P_{N,M,h}{F}^{-1},$$

and Theorem 5.1 of [20]. □

7. Proof of Universality

Theorems 2 and 3 are derived from Theorem 6 and Corollary 1, respectively, by using the Mergelyan theorem on the approximation of analytic functions by polynomials [23].

Proof of Theorem2. 

We recall that

$$S=\left\{g\in H(D):\mathrm{either}g(s)\ne 0\mathrm{for}\mathrm{all}s\in D,\mathrm{or}g(s)\equiv 0\right\},$$

It is well known, see, for example, [5], that the support of the measure $P_{\zeta }$ is the set S. Define the set

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

where $p(s)$ is a polynomial. Obviously, ${\mathrm{e}}^{p(s)}\in S$. Therefore, $G_{\epsilon }$ is an open neighbourhood of an element of the support of the measure $P_{\zeta }$. Thus, by a property of the support,

$$P_{\zeta }(G_{\epsilon })>0.$$

(21)

This, Theorem 6 and the equivalent of weak convergence in terms of open sets show that

$$\underset{N\to \infty }{lim\; inf}{P}_{N,M,h}(G_{\epsilon })⩾P_{\zeta }(G_{\epsilon })>0.$$

Hence, by the definition of $P_{N,M,h}$ and $G_{\epsilon }$,

$$\underset{N\to \infty }{lim\; inf}\frac{1}{M+1}\#\left\{N⩽k⩽N+M:\underset{s\in K}{sup}\left|\zeta (s+ih{\gamma }_k)-{\mathrm{e}}^{p(s)}\right|<\frac{\epsilon }{2}\right\}>0.$$

(22)

Now, we apply the Mergelyan theorem and choose the polynomial $p(s)$ satisfying

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

(23)

This and inequality (22) prove the first part of the theorem.

To prove the second part of the theorem, define the set

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

Then the set ${\widehat{G}}_{\epsilon }$ is a continuity set of the measure $P_{\zeta }$ for all but at most countably many $\epsilon >0$. This remark, Theorem 6 and the equivalent of weak convergence of probability measures in terms of open sets show that

$$\underset{N\to \infty }{lim}{P}_{N,M,h}({\widehat{G}}_{\epsilon })=P_{\zeta }({\widehat{G}}_{\epsilon })$$

(24)

for all but at most countably many $\epsilon >0$. Inequality (23) implies the inclusion $G_{\epsilon }\subset {\widehat{G}}_{\epsilon }$. Therefore, in view of inequality (21), we have $P_{\zeta }({\widehat{G}}_{\epsilon })>0$. This, Equation (24) and the definitions of $P_{N,M,h}$ and ${\widehat{G}}_{\epsilon }$ prove the second part of the theorem. □

Proof of Theorem 3. 

Denote by $S_F$ the support of the measure $P_{\zeta }{F}^{-1}$. We observe that $S_F$ contains the closure of the set $H_{a_1,\dots ,a_r;F}(D)$. Actually, let $g\in H_{a_1,\dots ,a_r;F}(D)$ and G be any open neighborhood of g. Then the set $F^{-1}G$ is open as well, and lies in S. Hence, $P_{\zeta }(F^{-1}G)>0$ because S is the support of $P_{\zeta }$. Therefore,

$$P_{\zeta }{F}^{-1}(G)=P_{\zeta }(F^{-1}G)>0.$$

This shows that $S_F$ contains the set $H_{a_1,\dots ,a_r;F}(D)$ and its closure.

Case $r=1$. By the Mergelyan theorem, there exists a polynomial $p(s)$ such that

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

(25)

Then, $p(s)\ne a_1$ for all $s\in K$ if $\epsilon$ is small enough. Therefore, by the Mergelyan theorem again, we find a polynomial $q(s)$ such that

$$\underset{s\in K}{sup}\left|(p(s)-a_1)-{\mathrm{e}}^{q(s)}\right|<\frac{\epsilon }{4}.$$

(26)

Since $g_1(s)\stackrel{def}{=}{\mathrm{e}}^{q(s)}+a_1\in H_{a_1;F}(D)$, the set

$${\mathcal{G}}_{\epsilon }=\left\{g\in H(D):\underset{s\in K}{sup}|g(s)-g_1(s)|<\frac{\epsilon }{2}\right\}$$

is an open subset of $S_F$. Hence,

$$P_{\zeta }{F}^{-1}({\mathcal{G}}_{\epsilon })>0.$$

(27)

This inequality together with Corollary 1, inequalities (25) and (26) prove the theorem in the case of the lower density.

In the case of density, consider the set ${\widehat{G}}_{\epsilon }$ defined in the proof of Theorem 2 which is a continuity set of the measure $P_{\zeta }{F}^{-1}$ for all but at most countably many $\epsilon >0$. Therefore, by Corollary 1,

$$\underset{N\to \infty }{lim}{P}_{N,M,h,F}({\widehat{G}}_{\epsilon })=P_{\zeta }{F}^{-1}({\widehat{G}}_{\epsilon }).$$

(28)

Inequalities (25) and (26) show that ${\mathcal{G}}_{\epsilon }\subset {\widehat{G}}_{\epsilon }$. Thus, by inequality (27), $P_{\zeta }{F}^{-1}({\widehat{G}}_{\epsilon })>0$. This, Equation (28) and the definitions of $P_{N,M,h,F}$ and ${\widehat{G}}_{\epsilon }$ prove the theorem in the case of density.

Case $r⩾2$. In this case, the function $f(s)$ lies in $S_F$. Therefore, the Mergelyan theorem is not needed, and the theorem follows immediately from Corollary 1. □