Une suite de matrices sym\'etriques en rapport avec la fonction de Mertens

TL;DR

This paper investigates symmetric matrices linked to the Mertens function, proposing a conjecture on their growth that could imply the Riemann hypothesis, highlighting the potential of matrix analysis in this area.

Contribution

It introduces a new class of symmetric matrices related to the Mertens function and suggests a conjecture connecting their properties to the Riemann hypothesis.

Findings

Numerical experiments support the conjecture.

The conjecture relates matrix norm growth to the Riemann hypothesis.

Matrix analysis may be key in understanding the Riemann hypothesis.

Abstract

In this paper we explore a class of equivalence relations over $\N^\ast$ from which is constructed a sequence of symetric matrices related to the Mertens function. From numerical experimentations we suggest a conjecture, about the growth of the quadratic norm of these matrices, which implies the Riemann hypothesis. This suggests that matrix analysis methods may play a more important part in this classical and difficult problem.