Refinements of Artin's primitive root conjecture

TL;DR

This paper refines Artin's primitive root conjecture by proposing new conjectures about the distribution of the order of integers modulo primes, proving some results under GRH and unconditionally.

Contribution

It introduces several refined conjectures on primitive roots, extending Artin's original conjecture, and proves these under GRH and in weaker unconditional forms.

Findings

Proposes new conjectures on the distribution of orders of integers mod p.

Proves these conjectures assuming the generalized Riemann hypothesis.

Establishes weaker unconditional results related to primitive roots.

Abstract

A famous conjecture of Artin asserts that any integer $a$ that is neither $-1$ nor a square should be a primitive root (mod $p$) for a positive proportion of primes $p$. Moreover, using a heuristic argument, Artin guessed an explicit formula for the proportion; this formula is well-supported by computations and is known to hold on a generalized Riemann hypothesis, but remains open. In this paper we propose several conjectures that capture the finer properties of the distribution of the order of $a$ (mod $p$) as $p$ varies over primes; these assertions contain Artin's original conjecture as a special case. We prove these conjectures assuming the generalized Riemann hypothesis, as well as weaker versions unconditionally.