Mean Values of Certain Multiplicative Functions and Artin's Conjecture on Primitive Roots

TL;DR

This paper explores a novel approach to Artin's conjecture on primitive roots by analyzing mean values of multiplicative functions and their Dirichlet series, offering new insights into the distribution of primitive roots.

Contribution

It introduces a method using Delange's theorem to study cyclotomic quantities and provides estimates for related Dirichlet series, advancing understanding of primitive roots.

Findings

New approach to Artin's conjecture via mean values of multiplicative functions

Estimates for the mean value of specific Dirichlet series

Insights into the distribution of primitive roots modulo primes

Abstract

We discuss how one could study asymptotics of cyclotomic quantities via the mean values of certain multiplicative functions and their Dirichlet series using a theorem of Delange. We show how this could provide a new approach to Artin's conjecture on primitive roots. We focus on whether a fixed prime has a certain order modulo infinitely many other primes. We also give an estimate for the mean value of one such Dirichlet series.