On good universality and the Riemann hypothesis

TL;DR

This paper employs ergodic theorems applied to specific transformations to provide new characterizations of the Riemann hypothesis and related conjectures, analyzing the value distribution of key zeta functions.

Contribution

It introduces novel ergodic-theoretic characterizations of the Riemann and Lindelöf hypotheses using subsequence and moving average theorems.

Findings

New ergodic characterizations of the Riemann hypothesis

Analysis of value distribution of Dirichlet L series and zeta functions

Extension of previous ergodic approaches to number theory

Abstract

We use subsequence and moving average ergodic theorems applied to Boole's transformation and its variants and their invariant measures on the real line to give new characterisations of the Lindelh{\"o}f Hypothesis and the Riemann hypothesis. These ideas are then used to study the value distribution of Dirichlet L series, and the zeta functions of Dedekind, Hurwitz and Riemann and their derivatives. This builds on earlier work of R. L. using Birkhoff's ergodic theorem and probability theory.