  In this paper, we show that Riemann hypothesis (concerning zeros of the zeta
function in the critical strip) is equivalent to the analytic continuation of
Euler products obtained by restricting the Euler zeta product to suitable
subsets $M_k$, $k\geq 1$ of the set of prime numbers. Each of these Euler
product defines so a partial zeta function $\zeta_{k}(s)$ equal to a Dirichlet
series of the form $\sum \epsilon(n)/n^s$, with coefficients $\epsilon(n)$
equal to 0 or 1 as n belongs or not to the population of integers generated by
$M_k$. We show that usual formulas of the arithmetic adapt themselves to such
populations (Moebius, Mertens, Lambert series,...). We envisage also the study
of summations inside these populations and new functions (generalizations of
the integer part function, of the harmonic series) directly connected to the
existence of analytical continuations.
