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Mathematics > Complex Variables
Title: Un exemple de somme de série de vecteurs propres à valeurs propres de module un, non récurrente
(Submitted on 8 Apr 2021 (v1), last revised 26 Apr 2024 (this version, v2))
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.
Submission history
From: Eric Saias [view email][v1] Thu, 8 Apr 2021 17:38:48 GMT (7kb)
[v2] Fri, 26 Apr 2024 07:32:43 GMT (0kb,I)
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