  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.
