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

TL;DR

This paper explores the probabilistic properties of Zeta laws in analytic number theory, analyzing coprime integer pairs and Gaussian integers, and revisiting fundamental Zeta function decompositions.

Contribution

It introduces the Zeta laws as a probabilistic framework for understanding integer coprimality and computes natural densities for coprime pairs in integers and Gaussian integers.

Findings

Computed natural density for coprime integer pairs

Extended the Zeta laws to Gaussian integers

Revisited the Euler product decomposition of the Zeta function

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.