Simple Proof of the Primitive Root Conjecture

TL;DR

This paper provides a simple, unconditional proof of the primitive root conjecture, establishing an asymptotic formula for counting primes where a fixed integer is a primitive root, improving upon previous conditional results.

Contribution

It offers a straightforward, unconditional proof of the primitive root conjecture's asymptotic formula, removing the need for unproven hypotheses.

Findings

Proves an unconditional asymptotic formula for primitive roots

Establishes a positive density constant elta(u)

Improves upon previous conditional results

Abstract

Let \(u\neq \pm 1,v^2\) be a fixed integer, let \(p\geq 2\) be a prime, and let $\text{ord}_p(u) \mid p-1$ be the multiplicative order of $u \text{ mod } p$. Define a prime counting function by $\pi(u,x)=\# \{ p\leq x:\text{ord}_p(u)=p-1 \}$. In 1967 Hooley proved a conditional asymptotic formula $\pi(u,x)=\delta(u)x(\log x)^{-1}+O(\log\log x(\log x)^{-2}$ for the primitive root conjecture. This note proves an unconditional asymptotic formula $\pi(u,x)=\delta(u)x(\log x)^{-1}+O(x(\log x)^{-2}$ of the same result, where $\delta(u)>0$ is the density constant.