Self-approximation of Dirichlet L-functions

TL;DR

This paper investigates the near self-approximation properties of Dirichlet L-functions for rational shifts, extending previous results for algebraic irrational and transcendental cases, and relates it to the Riemann hypothesis.

Contribution

It proves the self-approximation hypothesis for Dirichlet L-functions when the shift parameter is a nonzero rational number, filling a gap in the existing literature.

Findings

Established self-approximation for rational shifts in Dirichlet L-functions.

Connected the approximation property to the Riemann hypothesis for the case d=0.

Extended previous results from algebraic irrational and transcendental cases.

Abstract

Let $d$ be a real number, let $s$ be in a fixed compact set of the strip $1/2<\sigma<1$, and let $L(s, \chi)$ be the Dirichlet $L$-function. The hypothesis is that for any real number $d$ there exist 'many' real numbers $\tau$ such that the shifts $L(s+i\tau, \chi)$ and $L(s+id\tau, \chi)$ are 'near' each other. If $d$ is an algebraic irrational number then this was obtained by T. Nakamura. \L. Pa\'nkowski solved the case then $d$ is a transcendental number. We prove the case then $d\ne0$ is a rational number. If $d=0$ then by B. Bagchi we know that the above hypothesis is equivalent to the Riemann hypothesis for the given Dirichlet $L$-function. We also consider a more general version of the above problem.