Value-distribution of the Riemann zeta-function and related functions near the critical line

TL;DR

This paper investigates the value distribution of the Riemann zeta-function near the critical line, exploring universality, density of values, and implications for the Lindelöf hypothesis through advanced complex analysis techniques.

Contribution

It introduces a modified universality concept at the critical line, analyzes the density of zeta-values on approaching curves, and extends moment results to related Dirichlet series.

Findings

Critical line is a natural boundary for universality.

Values of zeta-function are dense along certain asymptotic curves.

Extended moment formulas support Lindelöf hypothesis implications.

Abstract

We study the value-distribution of the Riemann zeta-function and related functions on and near the critical line. Amongst others, we focus on the following: The critical line is a natural boundary of the Voronin-type universality property of the Riemann zeta-function. We study a modified concept of `universality' at the critical line: roughly speaking, we add a scaling factor to the vertical shifts that appear in Voronin's universality theorem. Levinson showed that almost all a-points of the Riemann zeta-function lie in a certain funnel-shaped region around the critical line. We rely on arguments of the theory of normal families and the notion of filling discs to detect a-points in this region which are very close to the critical line. According to a folklore conjecture (often attributed to Ramachandra) one expects that the values of the Riemann zeta-function on the critical line lie dense in the complex numbers. We show that there are certain curves which approach the critical line asymptotically such that the values of the zeta-function on these curves are dense in the complex numbers. The Lindel\"of hypothesis can be reformulated in terms of power moments to the right of the critical line. Tanaka showed recently that the expected asymptotic formulas for these power moments are true in a certain measure-theoretical sense. We provide a discrete and integrated version of Tanaka's result and extend it to a large class of Dirichlet series connected to the Riemann zeta-function.