Symmetric matrices related to the Mertens function

Abstract

In this paper we explore a family of congruences over ${\mathbb{N}}^{\ast }$ from which one builds a sequence of symmetric matrices related to the Mertens function. From the results of numerical experiments, we formulate a conjecture about the growth of the quadratic norm of these matrices, which implies the Riemann hypothesis. This suggests that matrix analysis methods may come to play a more important role in this classical and difficult problem.

Version submitted to LAA; some new references