A general inversion formula for summatory arithmetic functions and its application to the summatory function of the Moebius function

TL;DR

This paper introduces a new inversion formula for summatory arithmetic functions and applies it to relate the summatory Piltz divisor functions with the M"obius function, deriving results on the Riemann zeta-function.

Contribution

It presents a general inversion formula for summatory functions and demonstrates its application to connect divisor functions with the M"obius function, impacting number theory.

Findings

Derived bounds for $M(x,y)$ using divisor problem estimates

Established conditional results related to the zero-free region of the Riemann zeta-function

Connected summatory Piltz divisor functions with the M"obius function through the new inversion formula

Abstract

We prove an inversion formula for summatory arithmetic functions. As an application, we obtain an arithmetic relationship between summatory Piltz divisor functions and a sum of the M\"obius function over certain integers, denoted by $M(x,y)$. With this relationship, using bounds for the main and remainder terms in the $k$-divisor problems we deduce conditional and unconditional results concerning $M(x,y)$ and the zero-free region of the Riemann zeta-function and Dirichlet $L$-functions.