# Möbius function, an identity factory with applications

Date: Mon, Dec 4, 2023

Location: PIMS, University of British Columbia, Zoom, Online

Conference: Analytic Aspects of L-functions and Applications to Number Theory

Subject: Mathematics

Class: Scientific

CRG: L-Functions in Analytic Number Theory

### Abstract:

By using an identity relating a sum to an integral, we obtain a family of identities for the averages \(M(X)=\sum_{n\leq X} \mu(n)\) and \(m(X)=\sum_{n\leq X} \mu(n)/n\). Further, by choosing some specific families, we study two summatory functions related to the Möbius function, \(\mu(n)\), namely \(\sum_{n\leq X} \mu(n)/n^s\) and \(\sum{n\leq X} \mu(n)/n^s \log(X/n)\), where \(s\) is a complex number and \(\Re s >0\). We also explore some applications and examples when s is real. (joint work with O. Ramaré)