A Generalized Davenport Expansion

TL;DR

This paper generalizes Davenport's Fourier expansion for series involving fractional parts and introduces a Mellin transform linked to the Riemann zeta function, expanding analytical tools in number theory.

Contribution

It provides a new generalized Fourier expansion and a Mellin transform related to the Riemann zeta function, advancing analytical methods in number theory.

Findings

Generalized Davenport's Fourier expansion.

Established a Mellin transform connected to the Riemann zeta function.

Enhanced analytical techniques for series involving fractional parts.

Abstract

We prove a new generalization of Davenport's Fourier expansion of the infinite series involving the fractional part function over arithmetic functions. A new Mellin transform related to the Riemann zeta function is also established.