Primitive root biases for prime pairs I: existence and non-totality of biases

TL;DR

This paper investigates biases in primitive roots for prime pairs, proving under conjecture the existence of strong biases and showing that for some pairs, the expected inequality can be reversed.

Contribution

It establishes the existence of primitive root biases for prime pairs and demonstrates that these biases are not total, with some pairs exhibiting reversed inequalities.

Findings

Existence of strong sign biases under Bateman-Horn conjecture

Reversal of dominant inequalities for a positive proportion of prime pairs

Analysis of primitive root distributions for prime pairs

Abstract

We study the difference between the number of primitive roots modulo $p$ and modulo $p+k$ for prime pairs $p,p+k$. Assuming the Bateman-Horn conjecture, we prove the existence of strong sign biases for such pairs. More importantly, we prove that for a small positive proportion of prime pairs $p,p+k$, the dominant inequality is reversed.