Remarques \'el\'ementaires sur la fonction de M\"obius

TL;DR

This paper explores inequalities involving the M"obius function's summatory functions using M"obius inversion and Bernoulli polynomials, providing new insights into their relationships.

Contribution

It introduces novel inequalities between the logarithmic and ordinary summatory functions of the M"obius function using advanced mathematical tools.

Findings

Established inequalities between summatory functions of the M"obius function

Applied M"obius inversion and Bernoulli polynomials in new ways

Enhanced understanding of M"obius function properties

Abstract

We use M\"obius inversion and the Bernoulli polynomials to prove inequalities between the logarithmic summatory function of the M\"obius function and weighted averages of its ordinary summatory function.