Global Primitive Roots of Unity

TL;DR

This paper explores the properties of primitive roots of unity across infinite prime sets using ultraproducts, providing new insights into prime distribution and primitive root conjectures without relying on GRH.

Contribution

It introduces ultraproduct methods to resolve conjectures about primes with specific properties and extends results to the quantitative APRC via adelic torus measures.

Findings

Proves the infinitude of primes p where (p-1)/6 is prime.

Establishes the existence of primes p for which -1 is a primitive root.

Provides a GRH-free computation of the density of primes with a given primitive root property.

Abstract

An ideal setting to exhibit infinite sets of primes $p$ relative to which an integer is a primitive root $\pmod p$ is provided by the B\'ezout subdomain $\widetilde{\mathbb{B}}:=\mathbb{Z}^{\mathbb{P}}/\mathfrak{U}$ of the valuation domain $\widetilde{\mathbb{Z}}=\prod_{\mathfrak{U}} \mathbb{Z}_p$ with respect to a nonprincipal ultrafilter $\mathfrak{U}$ on $\mathbb{P}$, extant via Chebotarev's theorem and the ultrafilter theorem and such that the relative algebraic closure $\mathbb{L}:=\mathrm{Abs}(\widetilde{\mathbb{Q}})$ of the prime field of the valued field $\widetilde{\mathbb{Q}}=\prod_{\mathfrak{U}} \mathbb{Q}_p$ contains $\sqrt{-\tilde p}$ for $p\in\mathbb{P}$, contains no $\sqrt[3]{\tilde q}$ for $q\in\mathbb{P}$, and has $\mathrm{tor}(\mathbb{L}^\times)=\langle \zeta_6\rangle$. Results include positive resolutions of the conjectured infinitude of primes $p$ for which (i) $\frac{p-1}{6}$ is prime and (ii) a non-perfect-square $-1\neq m\in\mathbb{Z}$ is a primitive root $\pmod p$, establishing as manifest the efficacy of ultraproduct treatments in resolving number theory problems requiring certification of countably infinite conforming sets. Furthermore, we extend these results to the quantitative APRC via normalised ergodic Haar measure on the (monothetic) universal adelic torus $\mathrm{Hom}(\mathbb{Q}^{(\mathfrak{c})},\frac{\mathbb{R}}{\mathbb{Z}})$, leveraging B\'ezout rigidity of $\widetilde{\mathbb{B}}$ and the qualitative APRC witness set $T_m = \{ q\in\mathbb{P} \colon m\text{ is a primitive root}\!\pmod{q}\}$ to present a GRH-free computation of the natural density of $T_m$ as the corrected/entangled Artin Euler product $c_m\prod_{q\in\mathbb{P}}(1-\frac{1}{q(q-1)})$.