Un exemple de somme de série de vecteurs propres à valeurs propres de module un, non récurrente

Abstract

Let ${\zeta }^{\ast }(s)={\sum }_{n=1}^{+\infty }(-1{)}^n/n^s$ and $\tau$ the operator defined on the Frechet space of holomorphic functions in $\{s\in \mathbb{C}:1/2<Res<1\}$ by $\tau f(s)=f(s-2i\pi /\log 2)$. We show that the Riemann Hypothesis is equivalent to the strong recurrence of ${\zeta }^{\ast }(s)$ 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.

arXiv commentAfter submitting to arXiv, we found a trivial proof for the main result