Refinements of Artin's primitive root conjecture

TL;DR

This work refines Artin's primitive-root conjecture by encoding finer distributional properties of the ratio $(p-1)/ ext{ord}_p(a)$ across primes $p$ using generating functions for three statistics: $ ext{ω}$, $ ext{Ω}$, and $ ext{ω}- ext{ω}$, conditional on GRH. The authors extend Hooley's combinatorial framework to handle prime powers, connect the main terms to degrees of cyclotomic-Kummer extensions and Their Chebotarev densities, and derive explicit main-term Euler products with $a$-dependent correction factors. They provide unconditional bounds and a thorough numerical treatment, including recursive and non-recursive formulas for the associated density coefficients and their expectations, and they demonstrate how Artin's constant emerges as a base case when $z=0$. The methodology yields precise predictions for the distributions of these arithmetic statistics and offers both conditional (GRH) and unconditional results with potential for refinement and numerical verification. Overall, the paper delivers a comprehensive, quantified deformation of Artin-type densities that captures fine-grained statistical structure in the order of $a$ modulo primes and links it to explicit cyclotomic-Kummer field data.

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.

Figures (4)

• Figure 1: Coefficients and expectations for the "uncorrected" infinite products
• Figure 2: Coefficients and expectations for the case $a=3$
• Figure 3: Coefficients and expectations for the case $a=4$
• Figure 4: Coefficients and expectations for the case $a=5$

Theorems & Definitions (104)

• Definition 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Corollary 1.7
• Remark 1.8
• Theorem 1.10
• Corollary 1.11