Symmetric matrices related to the Mertens function

TL;DR

This paper investigates symmetric matrices derived from a family of congruences connected to the Mertens function, proposing a conjecture linking matrix norms to the Riemann hypothesis, highlighting potential new analytical approaches.

Contribution

It introduces a novel family of symmetric matrices related to the Mertens function and formulates a conjecture that connects their properties to the Riemann hypothesis.

Findings

Numerical experiments support the conjecture on matrix norm growth.

The conjecture implies the Riemann hypothesis.

Matrix analysis could be key in understanding the Mertens function.

Abstract

In 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.