Extention of the Laurin\v{c}ikas--Matsumoto theorem

A. Vaiginytė

Čebyševskij sbornik, Tome 20 (2019) no. 1, pp. 82-93.

Résumé

In 1975, S. M. Voronin discovered the remarkable universality property of the Riemann zeta-function $\zeta(s)$. He proved that analytic functions from a wide class can be approximated with a given accuracy by shifts $\zeta(s+i\tau)$, $\tau \in \mathbb{R},$ of one and the same function $\zeta(s)$. The Voronin discovery inspired to continue investigations in the field. It turned out that some other zeta and $L$-functions as well as certain classes of Dirichlet series are universal in the Voronin sense. Among them, Dirichlet $L$-functions, Dedekind, Hurwitz and Lerch zeta-functions. In 2001, A. Laurinčikas and K. Matsumoto obtained the universality of zeta-functions $\zeta(s, F)$ attached to certain cusp forms $F$. In the paper, the extention of the Laurinčikas-Matsumoto theorem is given by using the shifts $\zeta (s+i \varphi(\tau), F)$ for the approximation of analytic functions. Here $\varphi(\tau)$ is a differentiable real-valued positive increasing function, having, for $\tau \geqslant \tau_0,$ the monotonic continuous positive derivative, satisfying, for $\tau \rightarrow \infty,$ the conditions ${\frac{1}{\varphi'(\tau)}=o(\tau)}$ and $\varphi(2 \tau) \max_{\tau \leqslant t \leqslant 2\tau} \frac{1}{\varphi'(t)} \ll \tau$. More precisely, in the paper it is proved that, if $\kappa$ is the weight of the cusp form $F$, $K$ is the compact subset of the strip $\left\{s \in \mathbb{C}: \frac{\kappa}{2} \sigma \frac{\kappa+1}{2} \right\}$ with connected complement, and $f(s)$ is a continuous non-vanishing function on $K$ which is analytic in the interior of $K,$ then , for every $\varepsilon > 0,$ the set $\left\{\tau \in \mathbb{R}: \sup_{s \in K} | \zeta (s+i \varphi(\tau), F)-f(s) | \varepsilon \right\}$ has a positive lower density.

Détail

Keywords: zeta-function of cusp forms, Hecke-eigen cusp form, universality.

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