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This article is cited in 1 scientific paper (total in 1 paper) On the universality of the zeta functions of certain cusp forms A. Laurinčikas Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania
Russian version:
Matematicheskii Sbornik, 2022, Volume 213, Number 5, DOI: https://doi.org/10.4213/sm9650 
Bibliographic databases:
    § 1. IntroductionLet
$$
\begin{equation*}
\operatorname{SL}(2, \mathbb{Z})= \biggl\{\begin{pmatrix} a & b\\ c & d \end{pmatrix}\mkern-2mu\colon a,b,c,d\in \mathbb Z,\, ad-bc=1\biggr\}
\end{equation*}
\notag
$$
be the full modular group. An analytic function $\mathcal{F}(z)$ on the upper half-plane $\operatorname{Im} z>0$ that, for some $\kappa\in 2\mathbb{N}$ and all $\begin{pmatrix}a & b\\ c & d\end{pmatrix}\in \operatorname{SL}(2,\mathbb Z)$, satisfies the functional equation
$$
\begin{equation*}
\mathcal{F}\biggl(\frac{az+b}{cz+d}\biggr)= (cz+d)^\kappa \mathcal{F}(z),
\end{equation*}
\notag
$$
is called a modular form of weight $\kappa$. In this case $\mathcal{F}(z)$ has the following expansion in a Fourier series at infinity:
$$
\begin{equation*}
\mathcal{F}(z)=\sum_{m=-\infty}^\infty c(m) \mathrm{e}^{2\pi i m z}.
\end{equation*}
\notag
$$
If the Fourier coefficients $c(m)$ vanish for all $m\leqslant 0$, then $\mathcal{F}(z)$ is called a cusp form of weight $\kappa$. We assume in addition that the form $\mathcal{F}(z)$ is an eigenfunction of all Hecke operators
$$
\begin{equation*}
T_m \mathcal{F}(z)=m^{\kappa-1}\sum_{\substack{a,d>0 \\ ad=m}} \frac{1}{d^\kappa} \sum_{b({\mathrm{mod}}\, d)} \mathcal{F}\biggl(\frac{az+b}{d}\biggr), \qquad m\in \mathbb{N}.
\end{equation*}
\notag
$$
In this case $c(1)\neq 0$, and therefore, after a normalization, we can assume that $c(1)=1$. Thus, in what follows we assume that $\mathcal{F}(z)$ is a normalized Hecke cusp eigenform of weight $\kappa$. To study the Fourier coefficients $c(m)$ Hecke introduced the zeta function (or $L$-function) of the form
$$
\begin{equation*}
\zeta(s,\mathcal{F})=\sum_{m=1}^\infty \frac{c(m)}{m^s}, \qquad s=\sigma+it, \quad \sigma> \frac{\kappa+1}{2},
\end{equation*}
\notag
$$
which extends analytically to an entire function. Since the coefficients are multiplicative, the function $\zeta(s, \mathcal{F})$ factorizes into an Euler product over the primes in the half-plane $\sigma>(\kappa+1)/2$:
$$
\begin{equation*}
\zeta(s,\mathcal{F})= \prod_p \biggl(1-\frac{\alpha(p)}{p^s}\biggr)^{-1} \biggl(1-\frac{\beta(p)}{p^s}\biggr)^{-1},
\end{equation*}
\notag
$$
where $\alpha(p)$ and $\beta(p)$ are certain conjugate complex numbers such that ${\alpha(p) +\beta(p)= c(p)}$. As is well known, the function $\zeta(s, \mathcal{F})$ is universal in the sense of Voronin, who discovered the universality of the Riemann zeta function $\zeta(s)$ (see [1]). Roughly speaking, Voronin’s theorem states that any analytic function that has no zeros in the strip $\{s\in \mathbb C\colon 1/2<\sigma<1\}$ can be approximated by shifts $\zeta(s+i\tau)$, $\tau\in \mathbb R$, uniformly on compact subsets of this strip. Let $D_{\mathcal{F}}=\{s\in \mathbb C\colon \kappa/2<\sigma<(\kappa+1)/2\}$, let $\mathcal{K}_{\mathcal{F}}$ be the class of compact subsets with connected complement of $D_{\mathcal{F}}$, and $H_{0\mathcal{F}}(K)$, $K\in \mathcal{K}_{\mathcal{F}}$, be the class of continuous nonvanishing functions on $K$ that are analytic in the interior of $K$. Then the universality of the function $\zeta(s, \mathcal{F})$ is described by the following theorem (see [2]). Theorem 1. Let $K\in \mathcal{K}_{\mathcal{F}}$ and $f(s)\in H_{0\mathcal{F}}(K)$. Then for every $\varepsilon>0$ Here $\operatorname{meas} A$ stands for the Lebesgue measure of the measurable set $A\subset \mathbb R$. Theorem 1 is a universality theorem of continuous nature for $\zeta(s, \mathcal{F})$, because $\tau$ in $\zeta(s+i\tau, \mathcal{F})$ can take any values in the interval $[0,T]$. There is also a discrete version of Theorem 1. Let $\# A$ denote the cardinality of the set $A$. Let $h$ be a fixed positive number. Theorem 2 (see [3]). Let $K\in \mathcal{K}_{\mathcal{F}}$ and $f(s)\in H_{0\mathcal{F}}(K)$. Then for every $\varepsilon>0$ Here $N$ ranges over the set $\mathbb N_0=\mathbb N\cup\{0\}$. Theorems 1 and 2 show that there are infinitely many shifts of the form $\zeta\mkern-1mu(\mkern-2mu{s\!+\!i\tau}\mkern-1.5mu,\mkern-1.5mu \mathcal{F})$ and $\zeta(s+ikh, \mathcal{F})$, respectively, which approximate an arbitrary function in the class $H_{0\mathcal{F}}$. Unfortunately, none of the values of $\tau$ and $k$ is known. In this paper we prove that there exists a Dirichlet series which is absolutely convergent in the half-plane $\sigma> \kappa/2$ and depends on $T$ and $N$, respectively, such that the assertions of Theorems 1 and 2 hold for it. Let $\theta>0$ be a fixed number, let $u>0$, and let
$$
\begin{equation*}
v_u(m)= \exp\biggl\{-\biggl(\frac{m}{u}\biggr)^\theta\biggr\}, \qquad m\in \mathbb N.
\end{equation*}
\notag
$$
Consider the series
$$
\begin{equation*}
\zeta_u(s,\mathcal{F})= \sum_{m=1}^\infty \frac{c(m) v_u(m)}{m^s}.
\end{equation*}
\notag
$$
It follows from Mellin’s well-known formula
$$
\begin{equation*}
\frac{1}{2\pi i} \int_{b-i\infty}^{b+i\infty} \Gamma(z) a^{-z} \, \mathrm{d} z= \mathrm{e}^{-a}, \qquad a,b>0,
\end{equation*}
\notag
$$
where $\Gamma(z)$ is the Euler gamma function, that for every $A>0$ we have
$$
\begin{equation}
v_u(m)= \frac{1}{2\pi i} \int_{A\theta-i\infty}^{A\theta+i\infty} \frac{1}{\theta} \Gamma\biggl(\frac{z}{\theta}\biggr) \biggl(\frac{m}{u}\biggr)^{-z}\, \mathrm{d} z\ll_{u}m^{-A}.
\end{equation}
\tag{1.1}
$$
This bound, in combination with the bound
$$
\begin{equation}
c(m) \ll m^{\kappa/2-1/2+\varepsilon} \quad \forall\, \varepsilon>0,
\end{equation}
\tag{1.2}
$$
proves the absolute convergence of the series $\zeta_u(s, \mathcal{F})$ for any value of $\sigma$. Let $\mathcal{B}(\mathbb X)$ denote the Borel $\sigma$-field of the set $\mathbb X$ and $\mathbb P$ denote the set of all primes, and set $\gamma= \{ s\in \mathbb C\colon |s|=1\}$. Let where $\gamma_p=\gamma$ for all $p\in \mathbb P$. By Tychonoff’s theorem, the infinite-dimensional torus $\Omega$, with the product topology and the operation of pointwise multiplication, is a compact topological Abelian group. Therefore, the Haar probability measure $m_H$ can be defined on $(\Omega, \mathcal{B}(\Omega))$. We obtain a probability space $(\Omega, \mathcal{B}(\Omega), m_H)$. We denote the $p$th component, $p\in \mathbb P$, of an element $\omega\in \Omega$ by $\omega(p)$ and the space of analytic functions in $D_\mathcal{F}$ endowed with the topology of uniform convergence on compact subsets by $H(D_\mathcal{F})$. On the probability space $(\Omega, \mathcal{B}(\Omega), m_H)$ we consider the $H(D_\mathcal{F})$-valued random element
$$
\begin{equation*}
\zeta(s, \omega, \mathcal{F}) =\prod_{p\in \mathbb P} \biggl(1-\frac{\alpha(p)\omega(p)}{p^s} \biggr)^{-1} \biggl(1-\frac{\beta(p)\omega(p)}{p^s} \biggr)^{-1}.
\end{equation*}
\notag
$$
Let $P_{\zeta,\mathcal{F}}$ be the distribution of $\zeta(s, \omega, \mathcal{F})$, that is,
$$
\begin{equation*}
P_{\zeta, \mathcal{F}}(A)= m_H \bigl\{ \omega\in \Omega\colon \zeta(s, \omega, \mathcal{F})\in A\bigr\}, \qquad A\in \mathcal{B}(H(D_\mathcal{F})).
\end{equation*}
\notag
$$
Note that the product for $\zeta(s, \omega, \mathcal{F})$ converges for almost all $\omega\in \Omega$ uniformly on compact subsets of the strip $D_\mathcal{F}$. In this paper we prove the following universality theorems for the function $\zeta_u(s,\mathcal{F})$. Theorem 3. Assume that $u_T\,{\to}\, \infty$ as $T\,{\to}\,\infty$. Let $K\,{\in}\, \mathcal{K}_{\mathcal{F}}$ and ${f(s)\,{\in}\, H_{0\mathcal{F}}(K)}$. Then the following limit exists for all $\varepsilon>0$, apart from an at most countable set of values: The discrete case is more complicated due to the dependence on the arithmetic properties of the number $h$. We say that $h$ is a number of type 1 if for all $m\in \mathbb Z\setminus \{0\}$ the quantity $\exp\{ (2\pi m)/h\}$ is irrational. Otherwise $h$ is said to be a number of type 2. Let $\Omega_h$ be the closed subgroup of the group $\Omega$ generated by the element $(p^{-ih}\colon p\in \mathbb P)$. We extend $\omega(p)$, $p\in \mathbb P$, to the set $\mathbb N$ by the formula
$$
\begin{equation*}
\omega(m)= \mathop{\prod_{p^l\mid m}}_{p^{l+1}\nmid m} \omega^l(p), \qquad m\in \mathbb{N}.
\end{equation*}
\notag
$$
If $h$ is a number of type 2, then there is the least value $m_0\in \mathbb N$ such that $\exp\{(2\pi m_0)/h\}= a/b$ for $a,b\in \mathbb N$. As is known (see [4] and [3]),
$$
\begin{equation*}
\Omega_h= \begin{cases} \Omega & \text{if $h$ is of type 1}, \\ \{\omega\in \Omega\colon \omega(a)= \omega(b)\} & \text{if $h$ is of type 2}. \end{cases}
\end{equation*}
\notag
$$
Moreover, a Haar probability measure $m_H^h$ exists on $(\Omega_h, \mathcal{B}(\Omega_h))$, and the discrete universality theorem for the function $\zeta_u(s, \mathcal{F})$ has the following form. Theorem 4. Assume that $u_N\to \infty$ as $N\to\infty$ and let $h>0$ be a fixed number. Let $K\in \mathcal{K}_{\mathcal{F}}$ and $f(s)\in H_{0\mathcal{F}}(K)$. Then the following limit exists for all $\varepsilon>0$, apart from an at most countable set of values: The proofs of Theorems 3 and 4 use probabilistic limit theorems in the space of analytic functions, and also the closeness of the functions $\zeta_u(s,\mathcal{F})$ and $\zeta(s,\mathcal{F})$. § 2. Approximation in the meanWe recall the metric in the space $H(D_\mathcal{F})$. There is a sequence $\{K_l\colon l\in \mathbb N\}$ of compact subsets of the strip $D_\mathcal{F}$ such that $K_l \subset K_{l+1}$ for all $l\in \mathbb N$ and, if $K \subset D_\mathcal{F}$ is a compact set, then $K \subset K_l$ for some $l$. For example, we can take a sequence of embedded closed rectangles. Then
$$
\begin{equation*}
\rho(g_{1},g_{2})= \sum_{l=1}^{\infty}2^{-l}\frac{\sup_{s\in K_l}|g_{1}(s)-g_{2}(s)|}{1 +\sup_{s\in K_l}|g_{1}(s)-g_{2}(s)|}, \qquad g_1, g_2 \in H(D_\mathcal{F}),
\end{equation*}
\notag
$$
is a metric in $H(D_\mathcal{F})$ that induces the topology of uniform convergence on compact subsets. Proof. By the definition of the metric $\rho$, it suffices to prove the following equality for every compact set $K\subset D_\mathcal{F}$:
$$
\begin{equation*}
\lim_{T\to\infty} \frac{1}{T} \int_{0}^T \sup_{s\in K}\bigl|\zeta(s+i\tau, \mathcal{F})- \zeta_{u_T}(s+i\tau, \mathcal{F})\bigr|\, \mathrm{d} \tau =0.
\end{equation*}
\notag
$$
It follows from (1.1) that the following representation holds for $\alpha>1/2$: where
Let $K\subset D$ be a compact set. Then there is $\varepsilon>0$ such that $\kappa/2 +2\varepsilon\leqslant \sigma \leqslant (\kappa+1)/2-\varepsilon$ for all $s= \sigma+it\in K$. We set Then for all $s\in K$, (2.1) and the residue theorem imply that
$$
\begin{equation*}
\zeta_{u_T}(s+i\tau, \mathcal{F}) -\zeta(s+i\tau, \mathcal{F}) =\frac{1}{2\pi i} \int_{\theta_1-i\infty}^{\theta_1+i\infty} \zeta(s+z, \mathcal{F}) l_{u_T} (z) \,\frac{\mathrm{d} z}{z}.
\end{equation*}
\notag
$$
This entails that for all $s\in K$, replacing $t+v$ by $v$ we obtain
$$
\begin{equation*}
\begin{aligned} \, &\zeta_{u_T}(s+i\tau, \mathcal{F}) -\zeta(s+i\tau, \mathcal{F}) \\ &\qquad =\frac{1}{2\pi i} \int_{-\infty}^{\infty} \zeta\biggl(\frac{\kappa}{2}+\varepsilon +i\tau+iv+it, \mathcal{F}\biggr) \frac{l_{u_T}(\kappa/2+\varepsilon-\sigma+iv)}{\kappa/2+\varepsilon-s+iv}\, \mathrm{d} v \\ &\qquad \ll \int_{-\infty}^\infty \biggl|\zeta\biggl(\frac{\kappa}{2}+\varepsilon +i\tau+iv, \mathcal{F}\biggr)\biggr| \sup_{s\in K}\biggl|\frac{l_{u_T}(\kappa/2+\varepsilon-s+iv)}{\kappa/2 +\varepsilon-s+iv}\biggr|\, \mathrm{d} v. \end{aligned}
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
\begin{aligned} \, \notag &\frac{1}{T} \int_1^T \sup_{s\in K} \bigl| \zeta_{u_T}(s+i\tau, \mathcal{F}) -\zeta(s+i\tau, \mathcal{F})\bigr|\, \mathrm{d} \tau \\ &\ll \int_{-\infty}^{\infty}\biggl(\frac{1}{T} \int_1^T \biggl|\zeta\biggl(\frac{\kappa}{2}+\varepsilon +i\tau+iv, \mathcal{F}\biggr)\biggr|^2\, \mathrm{d} \tau\biggr)^{1/2} \sup_{s\in K}\biggl|\frac{l_{u_T}(\kappa/2+\varepsilon-s+iv)}{\kappa/2 +\varepsilon-s+iv}\biggr|\, \mathrm{d} v. \end{aligned}
\end{equation}
\tag{2.2}
$$
As is known, for $\kappa/2<\sigma<(\kappa+1)/2$ we have
$$
\begin{equation}
\int_0^T |\zeta(\sigma+it, \mathcal{F})|^2\, \mathrm{d} t\ll_\sigma T
\end{equation}
\tag{2.3}
$$
(see, for example, [5]). Hence for the same $\sigma$ and all $\tau\in \mathbb R$ we obtain
$$
\begin{equation}
\int_1^T |\zeta(\sigma+it+i\tau, \mathcal{F})|^2\, \mathrm{d} t\ll_\sigma T(1+|\tau|), \qquad T\to\infty.
\end{equation}
\tag{2.4}
$$
Using the classical bound which holds uniformly for $\sigma_1\leqslant \sigma\leqslant\sigma_2$ for arbitrary $\sigma_1<\sigma_2$, we see that for all ${\tau\,{\in}\, \mathbb R}$
$$
\begin{equation}
\begin{aligned} \, \notag \frac{l_{u_T}(\kappa/2+\varepsilon-s+iv)}{\kappa/2+\varepsilon-s+iv} &\ll_\theta u_T^{\kappa/2+\varepsilon-\sigma} \biggl|\Gamma\biggl( \frac{1}{\theta} \biggl(\frac{\kappa}{2} +\varepsilon-s+iv\biggr)\biggr) \biggr| \\ &\ll_\theta u_T^{-\varepsilon} \exp\biggl\{-\frac{c}{\theta}|v-\sigma|\biggr\} \ll_{\theta, K} u_T^{-\varepsilon} \exp\{ -c_1|v|\}. \end{aligned}
\end{equation}
\tag{2.5}
$$
In combination with (2.2) and (2.4) this proves that the right-hand side of (2.2) admits the bound which implies the assertion of the lemma. Let us now prove a discrete analogue of Lemma 1. Proof. Similarly to Lemma 1, it suffices to show that for any compact set $K\subset D_\mathcal{F}$ we have
$$
\begin{equation*}
\lim_{N\to\infty} \frac{1}{N+1} \sum_{k=0}^N \sup_{s\in K}\bigl|\zeta(s+ikh, \mathcal{F})- \zeta_{u_N}(s+ikh, \mathcal{F})\bigr| =0.
\end{equation*}
\notag
$$
Repeating the proof of Lemma 1 we see that
$$
\begin{equation}
\begin{aligned} \, \notag &\frac{1}{N+1} \sum_{k=0}^N \sup_{s\in K} \bigl| \zeta(s+ikh, \mathcal{F}) -\zeta_{u_N}(s+ikh, \mathcal{F})\bigr| \\ &\ll \int_{-\infty}^{\infty}\biggl(\frac{1}{N+1} \sum_{k=0}^N \biggl|\zeta\biggl(\frac{\kappa}{2}+\varepsilon +ikh+iv, \mathcal{F}\biggr)\biggr|^2 \biggr)^{1/2} \sup_{s\in K}\biggl|\frac{l_{u_N}(\kappa/2+\varepsilon-s+iv)}{\kappa/2 +\varepsilon-s+iv}\biggr|\, \mathrm{d} v. \end{aligned}
\end{equation}
\tag{2.6}
$$
Hence we need a bound for the discrete second moment of the function $\zeta(s, \mathcal{F})$. For this we use Gallagher’s lemma connecting the discrete and continuous second moments of some functions (see, for example, [6], Lemma 1.4). Thus,
$$
\begin{equation}
\begin{aligned} \, \notag &\sum_{k=2}^N \biggl|\zeta\biggl(\frac{\kappa}{2}+\varepsilon+ikh+iv, \mathcal{F}\biggr)\biggr|^2\ll_h \int_{0}^{Nh} \biggl|\zeta \biggl(\frac{\kappa}{2}+\varepsilon+i\tau+iv, \mathcal{F}\biggr)\biggr|^2 \, \mathrm{d} \tau \\ &\qquad +\biggl(\int_{0}^{Nh} \biggl|\zeta\biggl(\frac{\kappa}{2}+\varepsilon+i\tau+iv, \mathcal{F}\biggr)\biggr|^2 \, \mathrm{d} \tau\int_{0}^{Nh}\biggl|\zeta'\biggl(\frac{\kappa}{2}+\varepsilon+i\tau+iv, \mathcal{F}\biggr)\biggr|^2 \, \mathrm{d} \tau\biggr)^{1/2}. \end{aligned}
\end{equation}
\tag{2.7}
$$
It follows from (2.3) and Cauchy’s integral theorem that
$$
\begin{equation*}
\int_0^T |\zeta'(\sigma+ it, \mathcal{F})|^2\, \mathrm{d} t \ll_\sigma T
\end{equation*}
\notag
$$
for $\kappa/2<\sigma< (\kappa+1)/2$. Hence, for the same $\sigma$ and all $\tau\in \mathbb R$ we have
$$
\begin{equation*}
\int_0^T |\zeta'(\sigma+ it+i\tau, \mathcal{F})|^2\, \mathrm{d} t \ll_\sigma T(1+|\tau|), \qquad T\to\infty.
\end{equation*}
\notag
$$
From this, taking (2.4) and (2.7) into account we obtain
$$
\begin{equation}
\sum_{k=2}^N \biggl|\zeta\biggl(\frac{\kappa}{2}+\varepsilon+ikh+iv, \mathcal{F}\biggr)\biggr|^2\ll_{\varepsilon,h} N(1+|v|).
\end{equation}
\tag{2.8}
$$
From the approximate functional equation (see [7])
$$
\begin{equation*}
\zeta(s, \mathcal{F})= \sum_{m\leqslant (t/(2\pi))^2} \frac{c(m)}{m^s} +(-1)^{\kappa/2} (2\pi)^{2t-\kappa} \frac{\Gamma(\kappa-s)}{\Gamma(s)} +O(t^{\kappa-2\sigma}\log^2 t),
\end{equation*}
\notag
$$
where $|\sigma-\kappa/2|\leqslant 1/2$ and $t\geqslant t_0$, and from (1.2) we see that
$$
\begin{equation*}
\zeta(\sigma+it, \mathcal{F})\ll_\varepsilon |t|^{\kappa-2\sigma+1+\varepsilon}
\end{equation*}
\notag
$$
for $\kappa/2<\sigma< (\kappa+1)/2$. Therefore, for $\kappa/2<\sigma< (\kappa+1)/2$ we have
$$
\begin{equation*}
\sum_{k=0}^1 \biggl|\zeta\biggl(\frac{\kappa}{2}+\varepsilon+ikh+iv, \mathcal{F}\biggr)\biggr|^2\ll_{\varepsilon} 1+|v|^{2+\varepsilon}.
\end{equation*}
\notag
$$
Hence, taking (2.8), (2.5) and (2.6) into account we obtain the assertion of Lemma 2. § 3. Limit theoremsUsing limit theorems in the space $H(D_\mathcal{F})$ for the function $\zeta(s, \mathcal{F})$, we obtain similar theorems for $\zeta_u(s, \mathcal{F})$. For $A\in \mathcal{B}(H(D_\mathcal{F}))$ we set
$$
\begin{equation*}
P_{T, \mathcal{F}}(A)= \frac{1}{T} \operatorname{meas} \{\tau\in [0,T]\colon \zeta(s+i\tau, \mathcal{F})\in A\}.
\end{equation*}
\notag
$$
Lemma 3. $P_{T, \mathcal{F}}$ converges weakly to the measure $P_{\zeta, \mathcal{F}}$ as $T\to\infty$. Moreover, the support of $P_{\zeta, \mathcal{F}}$ is the set For the proof of the lemma, see [5]. We prove a similar assertion for
$$
\begin{equation*}
Q_{T, \mathcal{F}}(A)\stackrel{\mathrm{def}}{=} \frac{1}{T} \operatorname{meas} \{\tau\in [0,T]\colon \zeta_{u_T}(s+i\tau, \mathcal{F})\in A\}, \qquad A\in \mathcal{B}(H(D_\mathcal{F}).
\end{equation*}
\notag
$$
Theorem 5. Assume that $u_T\to\infty$ as $T\to\infty$. Then $Q_{T, \mathcal{F}}$ converges weakly to $P_{\zeta,\mathcal{F}}$ as $T\to\infty$. Proof. Let $\theta_T$ be a random variable defined on some probability space with measure $\mu$ and uniformly distributed on an interval $[0,T]$. Consider the $H(D_\mathcal{F})$-valued random elements and Then $P_{T,\mathcal{F}}$ and $Q_{T,\mathcal{F}}$ are the distributions of $X_{T,\mathcal{F}}$ and $Y_{T,\mathcal{F}}$, respectively.
We use the equivalent definition of the weak convergence of probability measures in terms of closed sets. Let $F\subset H(D_\mathcal{F})$ be a closed set. Then for $\varepsilon>0$ the set $F_\varepsilon=\{ {g\in H(D_\mathcal{F})}\colon \rho(g, F)\leqslant \varepsilon\}$ is also closed. Moreover,
$$
\begin{equation*}
\{Y_{T, \mathcal{F}}\in F\} \subset \{ X_{T, \mathcal{F}}\in F_\varepsilon\} \cup \{ \rho(X_{T,\mathcal{F}}, Y_{T, \mathcal{F}})\geqslant \varepsilon\}.
\end{equation*}
\notag
$$
This gives
$$
\begin{equation}
\mu\{ Y_{T, \mathcal{F}}\in F\} \leqslant \mu \{ X_{T,F}\in F_{\varepsilon}\} +\mu \{ \rho(X_{T,\mathcal{F}}, Y_{T, \mathcal{F}})\geqslant \varepsilon\}.
\end{equation}
\tag{3.1}
$$
It follows from Lemma 1 that
$$
\begin{equation}
\lim_{T\to\infty} \mu \{\rho(X_{T,\mathcal{F}}, Y_{T, \mathcal{F}})\geqslant \varepsilon\}\leqslant \lim_{T\to\infty} \frac{1}{T\varepsilon} \int_0^T \rho(\zeta(s+i\tau, \mathcal{F}), \zeta_{u_T}(s+i\tau, \mathcal{F}))\, \mathrm{d} \tau=0.
\end{equation}
\tag{3.2}
$$
Since
$$
\begin{equation*}
Q_{T, \mathcal{F}}(F) = \mu\{ Y_{T, \mathcal{F}}\in F\}\quad\text{and} \quad P_{T, \mathcal{F}}(F_\varepsilon) = \mu \{ X_{T,\mathcal{F}}\in F_\varepsilon\},
\end{equation*}
\notag
$$
it follows that (3.1) and (3.2) yield
$$
\begin{equation}
\limsup_{T\to\infty} Q_{T, \mathcal{F}}(F)\leqslant \limsup_{T\to\infty} P_{T, \mathcal{F}}(F_\varepsilon).
\end{equation}
\tag{3.3}
$$
However, by Lemma 3 and the equivalent definition of the weak convergence of probability measures in terms of closed sets (see [8], Theorem 2.1) we have
$$
\begin{equation*}
\limsup_{T\to\infty} P_{T, \mathcal{F}}(F_\varepsilon) \leqslant P_{\zeta, \mathcal{F}}(F_\varepsilon).
\end{equation*}
\notag
$$
Passing to the limit as $\varepsilon\to +0$ in this inequality and taking (3.3) into account, we obtain which means the weak convergence of $Q_{T, \mathcal{F}}$ to $P_{\zeta, \mathcal{F}}$. We denote elements of the set $\Omega_h$ by $\omega_h$ and their $p$th components by $\omega_h(p)$ and consider the $H(D_\mathcal{F})$-valued random element on the probability space $(\Omega_h, \mathcal{B}(\Omega_h), m_H^h)$
$$
\begin{equation*}
\zeta_h(s, \omega_h, \mathcal{F}) =\prod_{p\in \mathbb P} \biggl(1-\frac{\alpha(p) \omega_h(p)}{p^s}\biggr)^{-1} \biggl(1-\frac{\beta(p) \omega_h(p)}{p^s}\biggr)^{-1}.
\end{equation*}
\notag
$$
Let $P_{\zeta_h, \mathcal{F}}$ be the distribution of $\zeta_h(s, \omega_h, \mathcal{F})$, that is,
$$
\begin{equation*}
P_{\zeta_h, \mathcal{F}}(A) = m_H^h \{ \omega_h \in \Omega_h \colon \zeta_h(s, \omega_h, \mathcal{F})\in A\}, \qquad A\in \mathcal{B}(H(D_\mathcal{F})).
\end{equation*}
\notag
$$
For $A\in \mathcal{B}(H(D_\mathcal{F}))$ we set
$$
\begin{equation*}
P_{N, h, \mathcal{F}}(A) =\frac{1}{N+1} \# \{ 0\leqslant k\leqslant N\colon \zeta(s+ikh, \mathcal{F})\in A\}.
\end{equation*}
\notag
$$
The following assertion was proved in [3]. Lemma 4. $P_{N,h, \mathcal{F}}$ converges weakly to the measure $P_{\zeta_h, \mathcal{F}}$ as $N\to\infty$. Moreover, the support of $P_{\zeta_h, \mathcal{F}}$ is the set $S_\mathcal{F}$. Let
$$
\begin{equation*}
Q_{N,h, \mathcal{F}}(A)= \frac{1}{N+1} \# \{ 0\leqslant k\leqslant N\colon \zeta_{u_N}(s+ikh, \mathcal{F})\in A\}, \qquad A\in \mathcal{B}(H(D_\mathcal{F})).
\end{equation*}
\notag
$$
Then the following discrete limit theorem holds. Theorem 6. Assume that $u_N\to \infty$ as $N\to\infty$. Then $Q_{N,h, \mathcal{F}}$ converges weakly as $N\to\infty$ to the measure $P_{\zeta_h, \mathcal{F}}$. Proof. Consider a discrete random variable $\theta_{N,h}$ with distribution
$$
\begin{equation*}
\nu\{\theta_{N,h}= kh\}= \frac{1}{N+1}, \qquad k=0, 1, \dots, N,
\end{equation*}
\notag
$$
on a probability space with measure $\nu$ and the $H(D_\mathcal{F})$-valued random elements
$$
\begin{equation*}
X_{N,h, \mathcal{F}} = X_{N,h, \mathcal{F}}(s)= \zeta(s+i\theta_{N,h}, \mathcal{F})
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
Y_{N,h, \mathcal{F}} = Y_{N,h, \mathcal{F}}(s)= \zeta_{u_N}(s+i\theta_{N,h}, \mathcal{F}).
\end{equation*}
\notag
$$
Then $P_{N,h, \mathcal{F}}$ is the distribution of the element $X_{N,h, \mathcal{F}}$ and $Q_{N,h,\mathcal{F}}$ is the distribution of $Y_{N,h,\mathcal{F}}$. Let $F$ and $F_\varepsilon$ be the sets from the proof of Theorem 5. Then
$$
\begin{equation}
Q_{N,h, \mathcal{F}}(F)\leqslant P_{N,h, \mathcal{F}}(F_\varepsilon) +\nu\{ \rho(X_{N,h, \mathcal{F}}, Y_{N,h,\mathcal{F}})\geqslant\varepsilon\}.
\end{equation}
\tag{3.4}
$$
It follows from Lemma 2 that
$$
\begin{equation}
\begin{aligned} \, \notag &\lim_{N\to\infty} \nu(\{ \rho(X_{N, h, \mathcal{F}}, Y_{N, h, \mathcal{F}})\geqslant \varepsilon\} \\ &\qquad \leqslant \lim_{N\to\infty} \frac{1}{N+1} \sum_{k=0}^N \rho(\zeta(s+ikh, \mathcal{F}), \zeta_{u_N}(s+ikh, \mathcal{F}))=0. \end{aligned}
\end{equation}
\tag{3.5}
$$
Moreover, by Lemma 4 and the equivalent definition of the weak convergence of probability measures in terms of closed sets we have
$$
\begin{equation*}
\limsup_{N\to\infty} P_{N,h, \mathcal{F}}(F_\varepsilon)\leqslant P_{\zeta_h, \mathcal{F}}(F_\varepsilon).
\end{equation*}
\notag
$$
Thus, by (3.4) and (3.5) we have and taking the limit as $\varepsilon\to0$, we obtain the assertion of the lemma. § 4. Universality proofWe apply a useful property of the weak convergence of probability measures. Let $P$ be a probability measure on $(\mathbb X, \mathcal{B}(\mathbb X))$ and $v\colon \mathbb X\to \mathbb Y$ be a $(\mathcal{B}(\mathbb X), \mathcal{B}(\mathbb Y))$-measurable map. Then $P$ induces a unique probability measure $P v^{-1}$ on $(\mathbb Y, \mathcal{B}(\mathbb Y))$:
$$
\begin{equation*}
P v^{-1}(A)= P(v^{-1}A), \qquad A\in \mathcal{B}(\mathbb Y).
\end{equation*}
\notag
$$
The following assertion is well known (see, for example, [8]). Lemma 5. Let $P$ and $P_n$, $n\!\in\! \mathbb{N}$, be probability measures on $(\mathbb X, \mathcal{B}(\mathbb X))$ and $v\colon {\mathbb X\!\to\!\mathbb Y}$ be a continuous map. If $P_n$ converges weakly to $P$ as $n\to\infty$, then $P_nv^{-1}$ also converges weakly to $Pv^{-1}$ as $n\to\infty$. Proof of Theorem 3. The map $v\colon H(D_\mathcal{F})\to \mathbb R$ given by is continuous. Therefore, it follows from Theorem 5 and Lemma 5 that the measure
$$
\begin{equation*}
\frac{1}{T} \operatorname{meas}\Bigl\{ \tau\in [0,T]\colon \sup_{s\in K} |\zeta_{u_T}(s+i\tau, \mathcal{F})-f(s)|\in A\Bigr\}
\end{equation*}
\notag
$$
converges to
$$
\begin{equation*}
m_H\Bigl\{ \Omega\in \Omega\colon \sup_{s\in K} |\zeta(s, \omega, \mathcal{F})-f(s)|\in A\Bigr\}, \qquad A\in \mathcal{B}(\mathbb R),
\end{equation*}
\notag
$$
weakly as $T\to\infty$. As is well known, the weak convergence of probability measures on $(\mathbb R, \mathcal{B}(\mathbb R))$ is equivalent to the weak convergence of the corresponding distribution functions. Therefore, we see that the distribution function
$$
\begin{equation*}
\frac{1}{T} \operatorname{meas}\Bigl\{ \tau\in [0,T]\colon \sup_{s\in K} |\zeta_{u_T}(s+i\tau, \mathcal{F})-f(s)|<\varepsilon\Bigr\}
\end{equation*}
\notag
$$
converges weakly as $T\to\infty$ to the distribution function
$$
\begin{equation}
m_H\Bigl\{ \Omega\in \Omega\colon \sup_{s\in K} |\zeta(s, \omega, \mathcal{F})-f(s)|< \varepsilon\Bigr\}.
\end{equation}
\tag{4.1}
$$
Since the weak convergence of distribution functions is defined to be convergence to the limit function at all of its points of continuity, and the set of discontinuity points of a distribution function is at most countable, the existence of the limit (4.1) in the theorem is proved.
It remains to prove that the quantity (4.1) is positive. Since $f(s)\neq 0$ on $K$, it follows by Mergelyan’s theorem on the approximation of analytic functions by polynomials (see [9]) that there exists a polynomial $p(s)$ such that
$$
\begin{equation}
\sup_{s\in K} \bigl|f(s)- \mathrm{e}^{p(s)}\bigr|< \frac{\varepsilon}{2}.
\end{equation}
\tag{4.2}
$$
Then $\mathrm{e}^{p(s)}$ does not vanish and, by Lemma 3, this is an element of the support of the measure $P_{\zeta, \mathcal{F}}$. Hence, due to the properties of the support,
$$
\begin{equation}
m_H\biggl\{ \omega\in \Omega\colon \sup_{s\in K} \bigl| \zeta(s, \omega, \mathcal{F})-\mathrm{e}^{p(s)}\bigr|< \frac{\varepsilon}{2}\biggr\}>0.
\end{equation}
\tag{4.3}
$$
However, if $\omega \in \{ \omega \in \Omega\colon \sup_{s\in K} | \zeta(s, \omega, \mathcal{F})-\mathrm{e}^{p(s)}| < \varepsilon/2\}$, then $\omega\in \{ {\omega\in \Omega}$: $\sup_{s\in K}| \zeta(s, \omega, \mathcal{F})-f(s)|< \varepsilon\}$ by (4.2). Therefore, (4.3) and the monotonicity of the measure prove that (4.1) is positive.
This completes the proof of the theorem. Proof of Theorem 4. We apply the scheme used in the proof of Theorem 3. It follows from Theorem 6 and Lemma 5 that the measure
$$
\begin{equation*}
\frac{1}{N+1} \# \Bigl\{ 0\leqslant k\leqslant N\colon \sup_{s\in K} |\zeta_{u_N} (s+ikh, \mathcal{F}) -f(s)|\in A\Bigr\}
\end{equation*}
\notag
$$
converges weakly as $N\to\infty$ to
$$
\begin{equation*}
m_H^h\Bigl\{ \omega_h \in \Omega_h\colon \sup_{s\in K} |\zeta_h(s, \omega_h, \mathcal{F})-f(s)|\in A\Bigr\}, \qquad A\in \mathcal{B}(\mathbb R).
\end{equation*}
\notag
$$
This implies that the distribution function
$$
\begin{equation*}
\frac{1}{N+1} \# \Bigl\{ 0\leqslant k\leqslant N\colon \sup_{s\in K} |\zeta_{u_N} (s+ikh, \mathcal{F}) -f(s)|< \varepsilon\Bigr\}
\end{equation*}
\notag
$$
converges weakly as $N\to\infty$ to the distribution function
$$
\begin{equation}
m_H^h\Bigl\{ \omega_h \in \Omega_h\colon \sup_{s\in K} |\zeta_h(s, \omega_h, \mathcal{F})-f(s)|<\varepsilon\Bigr\}
\end{equation}
\tag{4.4}
$$
at all of its points of continuity. The positivity of the last function follows from Mergelyan’s theorem and Lemma 3.
This completes the proof of the theorem. Remark. Assume that $u_T\to\infty$ as $T \to \infty$ ($u_N \to \infty$ as $N \to \infty$). Let $K$ be a compact subset of the strip $D_\mathcal{F}$ and $f(s)$ be an analytic function without zeros in $D_\mathcal{F}$. Then the assertion of Theorem 3 (Theorem 4, respectively) holds. In fact, the condition $K\subset \mathcal{K}$ is used in Theorems 3 and 4 only to apply Mergelyan’s theorem. If the function $f(s)$ satisfies the conditions in the remark, then $f(s)\in S_\mathcal{F}$, and the positivity of (4.1) and (4.4) follows automatically from the properties of the support. 
Citation in format AMSBIB
\Bibitem{Lau22} |
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