Note on Artin's Conjecture on Primitive Roots

TL;DR

This paper explores a condition involving sums over primes related to the multiplicative order of integers, providing insights into Artin's conjecture on primitive roots and connecting it with cyclotomic periods.

Contribution

It relates the Murty-Srinivasan condition to sums involving cyclotomic periods, offering a new perspective on Artin's conjecture and its potential proofs.

Findings

Connected the Murty-Srinivasan condition to cyclotomic periods.

Provided a new framework linking primitive roots and cyclotomic subfields.

Suggested potential pathways for proving Artin's conjecture.

Abstract

E. Artin conjectured that any integer $a >1$ which is not a perfect square is a primitive root modulo $p$ for infinitely many primes $p.$ Let $f_a(p)$ be the multiplicative order of the non-square integer $a$ modulo the prime $p.$ M. R. Murty and S. Srinivasan [10] showed that if $\sum_{p<x} \frac 1 {f_a(p)}= O(x^{1/4})$ then Artin's conjecture is true for $a.$ We relate the Murty-Srinivasan condition to sums involving the cyclotomic periods from the subfields of $\mathbb Q(e^{2{\pi}i/p})$ corresponding to the subgroups $<a> \subseteq \mathbb F*_p.$