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arXiv · 1707.06517

Simple Proof of the Primitive Root Conjecture

N. A. Carella
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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.

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N. A. Carella. 2017-07-14. Simple Proof of the Primitive Root Conjecture. https://arxiv.org/abs/1707.06517

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