Average $r$-rank Artin Conjecture

TL;DR

This paper establishes asymptotic formulas for the average number of primes with a given index in the reduction of finitely generated subgroups of rational numbers, extending classical Artin conjecture results.

Contribution

It provides new asymptotic formulas for the average and mean square error of primes related to subgroup indices, with uniformity ranges.

Findings

Asymptotic formula for the average number of primes with a specified index.

Asymptotic formula for the mean square of the error terms.

Extension of classical Artin conjecture to higher ranks and subgroup structures.

Abstract

Let $\Gamma\subset\mathbb{Q}^*$ be a finitely generated subgroup and let $p$ be a prime such that the reduction group $\Gamma_p$ is a well defined subgroup of the multiplicative group $\mathbb{F}_p^*$. We prove an asymptotic formula for the average of the number of primes $p\le x$ for which the index $[\mathbb{F}_p^*:\Gamma_p]=m$. The average is performed over all finitely generated subgroups $\Gamma=\langle a_1,\dots,a_r \rangle\subset\mathbb{Q}^*$, with $a_i\in\mathbb{Z}$ and $a_i\le T_i$ with a range of uniformity: $T_i>\exp(4(\log x \log\log x)^{\frac{1}{2}})$ for every $i=1,\dots,r$. We also prove an asymptotic formula for the mean square of the error terms in the asymptotic formula with a similar range of uniformity. The case of rank $1$ and $m=1$ corresponds to the classical Artin conjecture for primitive roots and has already been considered by Stephens in 1969.