Recurrence in topological dynamics and the Riemann hypothesis (Q1100501)

In this paper the Riemann hypothesis is reformulated in terms of topological dynamics. The paper is a continuation of the author's doctoral thesis. To give the main result, we need some definitions. A Dirichlet character modulo k is an arithmetic function nonvanishing only for integers co- prime to k which is completely multiplicative and periodic. Corresponding to each Dirichlet character \(\chi\), one defines the Dirichlet L-function L(.,\(\chi)\) by the series \(L(z,\chi)=\sum^{\infty}_{n=1}\chi (n)n^{-z}\). For \(k=1\) one obtains \(\chi =\chi_ 0\) and \(L(.,\chi_ 0)=\zeta\) the zeta-function. The Riemann hypothesis for the character \(\chi\) conjectures that the nontrivial zeros of L(.,\(\chi)\) lie on the critical line \(Re(z)=.\) Denoting by \(\Omega\) the set of complex numbers z, such that \(<Re(z)<1\) and by H the space of all analytic functions on \(\Omega\), with the topology of uniform convergence on compacta, one defines the flow (H,\({\mathbb{R}},\sigma)\) by \(\sigma (f,t)=g\) where \(g(z)=f(z+it)\). It is proved that the Riemann hypothesis holds for L(.,\(\chi)\) if and only if L(.,\(\chi)\) is a strongly recurrent point of the flow (H,\({\mathbb{R}},\sigma)\).