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arXiv · 1301.4202

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

Sergei Preobrazhenskii
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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 Möbius 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.

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Sergei Preobrazhenskii. 2013-10-10. A general inversion formula for summatory arithmetic functions and its application to the summatory function of the Moebius function. https://arxiv.org/abs/1301.4202

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