Universality of some functions related to zeta-functions of certain cusp forms

Abstract

It is well-known that the zeta-function $\zeta(s,F)$ attached to a normalized Hecke-eigen cusp form $F$ of weight $\kappa$ is universal in the sense that their shifts $\zeta(s + i\tau,F)$ with appropriate $\tau \in \mathbb{R}$ approximate any analytic function uniformly on compact subsets of the strip $\{s = \sigma+it \in \mathbb{C}\colon \kappa/2 < \sigma < (\kappa+1)/2\}$ with any prescribed accuracy. In this paper we consider some classes of operators $\Phi$ such that the function $\Phi(\zeta(s,F))$ is universal in the above sense. In particular, this implies the universality of the functions, for example, $\zeta(s,F)^{N}$ ($N$-th power) and $\zeta^{(N)}(s,F)$ ($N$-th derivative) with $N \in \mathbb{N}$, $e^{\zeta(s,F)}$, $\sin\zeta(s,F)$, and $\cos\zeta(s,F)$.