On the Mean Value of a Weighted Composite Arithmetic 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.