Harmonic Analysis of some arithmetical functions

TL;DR

This paper explores the harmonic analysis of arithmetical functions through their power and Dirichlet series representations, linking them to Hilbert spaces and employing key estimates and operations to deepen understanding.

Contribution

It introduces a novel operation on power series and connects arithmetical functions to Hilbert spaces, expanding analytical tools in number theory.

Findings

Established bounds using Davenport's estimate on M"obius sums

Linked arithmetical functions to Hilbert space frameworks

Developed an operation on power series related to these functions

Abstract

We study three functions which are power series in the variable $z$, Dirichlet series in the variable $s$ and with coefficients given by arithmetical functions. A strong point is to relate these functions to some Hilbert spaces. Three main ingredients are used: an estimate of Davenport on sums of M\"obius functions, a result of Lucht on convolutions of arithmetical Dirichlet series and the introduction of an operation $\otimes$ on power series, naturally associated with the mentioned Hilbert spaces.