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

## TL;DR

This paper establishes a mean value theorem for a class of density-like arithmetic functions defined by products over divisors, with applications to densities of normal and primitive elements in finite field extensions.

## Contribution

It introduces a mean value theorem for functions of the form f(n)=∏_{d|n}g(d), expanding understanding of their average behavior under certain conditions.

## Key findings

- Derived mean value formulas for the class of functions
- Applied results to densities of normal elements in finite fields
- Connected density functions to their mean values

## 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.