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Let $\zeta^*(s)=\sum_{n=1}^{+\infty}(-1)^n/n^s$ and $\tau$ the operatordefined 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 RiemannHypothesis 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 aseries of eigenvectors with unimodular eigenvalues for an operator $u$ isstrongly recurrent for $u$. But we give a counterexample showing that it is notthe case.

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