2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/222485In this paper we explore a family of congruences over $\N^\ast$ 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 referencesNumber Theory11C20; 11N56; 15A36; 15A60text