On joint approximation of analytic functions by nonlinear shifts of zeta-functions of certain cusp forms

Antanas Laurinčikas,Darius Šiaučiūnas,Adelė Vaiginytė

doi:10.15388/namc.2020.25.15734

Antanas Laurinčikas, Darius Šiaučiūnas + Show 1 more

Open Access

https://doi.org/10.15388/namc.2020.25.15734

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Abstract

In the paper, joint discrete universality theorems on the simultaneous approximation of a collection of analytic functions by a collection of discrete shifts of zeta-functions attached to normalized Hecke-eigen cusp forms are obtained. These shifts are defined by means of nonlinear differentiable functions that satisfy certain growth conditions, and their combination on positive integers is uniformly distributed modulo 1.

Highlights

It is known that some of zeta and L-functions are universal in the sense that their shifts approximate wide classes of analytic functions
In [21], he proved a joint universality theorem for Dirichlet L-functions L(s, χ1), . . . , L(s, χr) with nonequivalent Dirichlet characters, see [2, 4, 7]
Nakamura and Pankowski proved in [14] the joint universality theorem for arbitrary number of automorphic zeta-functions

Summary

Introduction

It is known that some of zeta and L-functions are universal in the sense that their shifts approximate wide classes of analytic functions. The zeta-function ζ(s, F ) of the cusp form F (z) is defined for σ > (κ + 1)/2 by the Dirichlet series. In [21], he proved a joint universality theorem for Dirichlet L-functions L(s, χ1), . The zeta-functions approximating a collection of analytic functions must be independent in a certain sense. If the coefficients of Dirichlet series defining zeta-functions are nonperiodic, the problem of joint universality for those zeta-functions becomes very complicated This remark concerns the zeta-functions of cusp forms. Nakamura and Pankowski proved in [14] the joint universality theorem for arbitrary number of automorphic zeta-functions. In [8], joint discrete universality theorems for zeta-functions of cusp forms were obtained. Involving the uniform distribution modulo 1 makes the probabilistic method very convenient for the proof of universality theorems

Probabilistic model

Proof of universality theorems