On the Value-Distribution of Hurwitz Zeta-Functions with Algebraic Parameter

TL;DR

This paper investigates the value distribution of Hurwitz zeta-functions with algebraic irrational parameters, demonstrating effective denseness in certain strips and providing initial evidence of universality properties.

Contribution

It establishes effective denseness results for Hurwitz zeta-functions with algebraic irrational parameters, a first step towards understanding their universality.

Findings

Proves effective denseness of Hurwitz zeta-function values and derivatives

Shows results in strips near the critical line

Provides initial evidence of universality for these zeta-functions

Abstract

We study the value-distribution of the Hurwitz zeta-function with algebraic irrational parameter $\zeta(s;\alpha)=\sum_{n\geq_0}(n+\alpha)^{-s}$. In particular, we prove effective denseness results of the Hurwitz zeta-function and its derivatives in suitable strips containing the right boundary of the critical strip $1+i\mathbb{R}$. This may be considered as a first "weak" manifestation of universality for those zeta-functions.