Mean value theorems for a class of density-like arithmetic functions

Abstract

This paper provides a mean value theorem for arithmetic functions $f$ defined by $$f(n)=\prod _{d|n}g(d),$$ where $g$ is an arithmetic function taking values in $(0,1]$ and satisfying some generic conditions. As an application of our main result, we prove that the density ${\mu }_q(n)$ (resp. ${\rho }_q(n)$) of normal (resp. primitive) elements in the finite field extension ${\mathbb{F}}_{q^n}$ of ${\mathbb{F}}_q$ are arithmetic functions of (non zero) mean values.

To appear in IJNT