Un exemple de somme de s\'erie de vecteurs propres \`a valeurs propres de module un, non r\'ecurrente

TL;DR

This paper explores the connection between the Riemann Hypothesis and the recurrence properties of a specific series related to the zeta function, providing insights into eigenvector sums and counterexamples.

Contribution

It establishes the equivalence of the Riemann Hypothesis with the strong recurrence of a particular series under a shift operator and discusses the limitations of eigenvector sum recurrence.

Findings

Riemann Hypothesis is equivalent to the strong recurrence of ^*(s) under pi/log 2 shift

A sufficient condition for RH involves eigenvector sum recurrence, which is shown to be false in general

Counterexample demonstrates that eigenvector sum recurrence does not always imply strong recurrence

Abstract

Let $\zeta^*(s)=\sum_{n=1}^{+\infty}(-1)^n/n^s$ and $\tau$ the operator defined on the Frechet space of holomorphic functions in $\{s\in \mathbb C :1/2< Re \, s<1\}$ by $\tau f(s)= f(s-2i\pi/\log 2)$. We show that the Riemann Hypothesis is equivalent to the strong recurrence of $\zeta^*(s)$ for $\tau$. It follows that a sufficient condition for $RH$ would be that every sum of a series of eigenvectors with unimodular eigenvalues for an operator $u$ is strongly recurrent for $u$. But we give a counterexample showing that it is not the case.