Partial Euler products as a new approach to Riemann hypothesis

Abstract

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.