Joint Universality of the Zeta-Functions of Cusp Forms

Renata Macaitienė

doi:10.3390/math9172161

Abstract

Let F be the normalized Hecke-eigen cusp form for the full modular group and ζ(s,F) be the corresponding zeta-function. In the paper, the joint universality theorem on the approximation of a collection of analytic functions by shifts (ζ(s+ih1τ,F),…,ζ(s+ihrτ,F)) is proved. Here, h1,…,hr are algebraic numbers linearly independent over the field of rational numbers.

Highlights

The series of the types∑ ∑ ∞ am m=1 ms and ∞ame−λms, m=1 s = σ + it, Citation: Macaitiene, R
This paper is devoted to the universality of zeta-functions of certain cusp forms
In [7], a universality theorem was obtained for zeta-functions of new forms

Summary

The series of the types

Joint Universality of the Zeta-Functions of Cusp Forms. This paper is devoted to the universality of zeta-functions of certain cusp forms. In [7], a universality theorem was obtained for zeta-functions of new forms. The first result in this direction belongs to Voronin He considered [11] the functional independence of Dirichlet L-functions L(s, χ) with pairwise nonequivalent Dirichlet characters χ and, for this, he obtained their joint universality. The paper [12] is devoted to the joint universality for zeta-functions of new forms twisted by Dirichlet characters, i.e., for the functions m=1 c(m)χ(m) ms. Continuous and discrete joint universality theorems for more general zeta-functions are given in [14,15,16]. Our aim is to obtain a joint universality theorem for zeta-functions of normalized Hecke-eigen cusp forms by using different shifts. Ρ is a metric in Hr(Dκ) inducing the product topology

Let θ

Findings

Define the set