Near-primitive roots

TL;DR

This paper computes the density of primes for which a given rational number is a near-primitive root of a specified index, extending previous work and confirming recent algebraic results under GRH.

Contribution

It explicitly determines the density of near-primitive roots of a given index for rational numbers, generalizing prior cases and aligning with recent algebraic density calculations.

Findings

Density expressed as g(g)A under GRH

Explicit formula for the density involving g(g) and Artin constant

Agreement with recent algebraic density results

Abstract

Given an integer $t\ge 1$, a rational number $g$ and a prime $p\equiv 1({\rm mod} t)$ we say that $g$ is a near-primitive root of index $t$ if $\nu_p(g)=0$, and $g$ is of order $(p-1)/t$ modulo $p$. In the case $g$ is not minus a square we compute the density, under the Generalized Riemann Hypothesis (GRH), of such primes explicitly in the form $\rho(g)A$, with $\rho(g)$ a rational number and $A$ the Artin constant. We follow in this the approach of Wagstaff, who had dealt earlier with the case where $g$ is not minus a square. The outcome is in complete agreement with the recent determination of the density using a very different, much more algebraic, approach due to Hendrik Lenstra, the author and Peter Stevenhagen.