Discrete universality in the Selberg class
In [Izv. Akad. Nauk SSSR, Ser. Mat. 39, 475--486 (1975; Zbl 0315.10037)], \textit{S. M. Voronin} discovered a universality property of the Riemann zeta-function \(\zeta(s)\) in the strip \(\{s\in\mathbb{C}:1/2<\mathrm{Re}(s)<1\}\), which can be stated in the following modern version: let \(K\) be a compact subset of \(\{s\in\mathbb{C}:1/2<\mathrm{Re} s<1\}\) with connected complement and let \(f(s)\) be a non-vanishing continuous function on K which is analytic in the interior of \(K\). Then, for every \(\varepsilon>0\), we have \[ \liminf_{T\to\infty}\frac{1}{T}\ \mu\left\{\tau\in[0,T]: \max_{s\in K}|\zeta(s+i\tau)-f(s)|<\varepsilon\right\}>0, \] where \(\mu(\cdot)\) denotes the Lebesgue measure. It is also known that the universality theorem holds for other \(L\)-functions important in analytic number theory. A quite general result in this direction was proved by \textit{H. Nagoshi} and \textit{J. Steuding} [Lith. Math. J. 50, No. 3, 293--311 (2010; Zbl 1241.11109)], who showed universality in the strip \(\{s\in\mathbb{C}:\max(1/2,1-1/d_L)<\mathrm{Re}(s)<1\}\) (\(d_L\) denotes the degree of a given \(L\)-function) for all elements \(L(s)=\sum_{n=1}^\infty a(n)n^{-s}\) from the Selberg class, provided \(\sum_{p\leq x}|a(p)|^2\sim \kappa \pi(x)\) for some \(\kappa>0\), where \(\pi(x)\) counts all primes not exceeding \(x\). On the other hand, it is known that the discrete version of universality also holds for the Riemann zeta-function and its various generalizations; here discrete means that the shifts \(\tau\) are taken from an arithmetic progression. The main result of the paper under review states that the discrete universality theorem in the strip \(\{s\in\mathbb{C}:\max(1/2,1-1/d_L)<\mathrm{Re}(s)<1\}\) holds also for every \(L\)-function \(L(s)=\sum_{n=1}^\infty a(n)n^{-s}\) from the Selberg class satisfying \(\sum_{p\leq x}|a(p)|^2\sim \kappa \pi(x)\) for some \(\kappa>0\).
scientific article; zbMATH DE number 6919530
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