On the Mean Value of a Weighted Composite Arithmetic Function

TL;DR

This paper uses analytic number theory to derive an asymptotic formula for the mean value of a weighted composite arithmetic function involving the divisor function and a minimal power function.

Contribution

It introduces a novel approach to analyze the mean values of weighted composite functions using Dirichlet series and contour integration techniques.

Findings

Established a rigorous asymptotic formula for the weighted sum

Analyzed the analytic properties of the associated Dirichlet series

Provided insights into the distribution of the composite function

Abstract

The primary objective of this paper is to employ methods from analytic number theory to investigate the mean value properties of a composite function involving the Dirichlet divisor function and a generalized minimal power function. Specifically, we study the weighted summatory function where the divisor function is normalized by the number of distinct prime factors. We establish a rigorous asymptotic formula for this sum, detailing the analytic properties of the associated Dirichlet series and the contour integration process.