On a basic mean value Theorem with explicit exponents

TL;DR

This paper enhances and completes a general proof of Vaughan's basic mean value theorem for a class of arithmetic functions, extending its application to generalized von Mangoldt functions.

Contribution

It provides a more comprehensive and explicit version of the mean value theorem with improved exponents for certain arithmetic functions.

Findings

Derived a mean value theorem for generalized von Mangoldt functions.

Extended previous results to a broader class of arithmetic functions.

Improved the explicit exponents in the mean value estimates.

Abstract

In this paper we follow a paper from A. Sedunova (2017) regarding R. C. Vaughan's basic mean value Theorem (Acta Arith. 1980) to improve and complete a more general demonstration for a suitable class of arithmetic functions as started by A. C. Cojocaru and M. R. Murty (London Math. Soc. Stud. Texts 2006). As an application we derive a basic mean value Theorem for the von Mangoldt generalized functions.