Joint universality of periodic zeta-functions with multiplicative coefficients. II

Antanas Laurinčikas,Monika Tekorė,Darius Šiaučiūnas

doi:10.15388/namc.2021.26.23934

Antanas Laurinčikas, Monika Tekorė + Show 1 more

Open Access

https://doi.org/10.15388/namc.2021.26.23934

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Abstract

In the paper, a joint discrete universality theorem for periodic zeta-functions with multiplicative coefficients on the approximation of analytic functions by shifts involving the sequence f kg of imaginary parts of nontrivial zeros of the Riemann zeta-function is obtained. For its proof, a weak form of the Montgomery pair correlation conjecture is used. The paper is a continuation of [A. Laurinčikas, M. Tekorė, Joint universality of periodic zeta-functions with multiplicative coefficients, Nonlinear Anal. Model. Control, 25(5):860–883, 2020] using nonlinear shifts for approximation of analytic functions.

Highlights

It is well known that some zeta- and L-functions, and even some classes of Dirichlet series, for example, the Selberg-Steuding class, see [29, 32], are universal in the Voronin sense, i.e., a wide class of analytic functions can be approximated by one and the same zeta-function
We focus on joint universality of socalled periodic zeta-functions with generalized shifts involving the sequence {γk: k ∈ N} of imaginary parts of nontrivial zeros of the function ζ(s)
In [21], joint continuous universality theorems for periodic zeta-functions with shifts defined by means of certain differentiable functions were obtained

Summary

Introduction

It is well known that some zeta- and L-functions, and even some classes of Dirichlet series, for example, the Selberg-Steuding class, see [29, 32], are universal in the Voronin sense, i.e., a wide class of analytic functions can be approximated by one and the same zeta-function. In [5], the shifts ζ(s + ihγk) were applied, where {γk: k ∈ N} = {γk: 0 < γ1 < · · · γk γk+1 · · · } is the sequence of imaginary parts of nontrivial zeros of the Riemann zeta-function. Joint universality of zeta-functions was widely studied, and many results are known; see, for example, general results obtained in [7,8,9,10,11,14,26,30] and other papers by authors of the mentioned works. We focus on joint universality of socalled periodic zeta-functions with generalized shifts involving the sequence {γk: k ∈ N} of imaginary parts of nontrivial zeros of the function ζ(s). In [21], joint continuous universality theorems for periodic zeta-functions with shifts defined by means of certain differentiable functions were obtained.

Limit theorems

Findings

Proof of Theorem 2