Les lois Z\^eta pour l'arithm\'etique

Abstract

This paper provides a probabilist point of view about some results in analytic number theory. The main tool is the family of Zeta laws, which is a consolation for the non-existence of an uniform law on the set of integers. We prove the existence and compute the natural density for the pairs of coprime integers, and also for the pairs of coprime Gaussian integers.Along the way, we recover the decomposition of the Zeta function as an Eulerian product and some related results.