  $L$ functions based on Dirichlet characters are natural generalizations of
the Riemann $\zeta(s)$ function: they both have series representations and
satisfy an Euler product representation, i.e. an infinite product taken over
prime numbers. In this paper we address the Generalized Riemann Hypothesis
relative to the non-trivial complex zeros of the Dirichlet $L$ functions by
studying the possibility to enlarge the original domain of convergence of their
Euler product. The feasibility of this analytic continuation is ruled by the
asymptotic behavior in $N$ of the series $B_N = \sum_{n=1}^N \cos \bigl( t \log
p_n - \arg \chi (p_n) \bigr)$ involving Dirichlet characters $\chi$ modulo $q$
on primes $p_n$. Although deterministic, these series have pronounced
stochastic features which make them analogous to random time series. We show
that the $B_N$'s satisfy various normal law probability distributions. The
study of their large asymptotic behavior poses an interesting problem of
statistical physics equivalent to the Single Brownian Trajectory Problem, here
addressed by defining an appropriate ensemble $\mathcal{E}$ involving intervals
of primes. For non-principal characters, we show that the series $B_N$ present
a universal diffusive random walk behavior $B_N = O(\sqrt{N})$ in view of the
Dirichlet theorem on the equidistribution of reduced residue classes modulo $q$
and the Lemke Oliver-Soundararajan conjecture on the distribution of pairs of
residues on consecutive primes. This purely diffusive behavior of $B_N$ implies
that the domain of convergence of the infinite product representation of the
Dirichlet $L$-functions for non-principal characters can be extended from
$\Re(s) > 1$ down to $\Re (s) = \frac{1}{2}$, without encountering any zeros
before reaching this critical line.
