On the Second Moment Estimate Involving the $\lambda$-Primitive Roots Modulo $n$

TL;DR

This paper extends the analysis of second moment estimates for primitive roots to composite moduli, providing new average results and filling a gap in the existing literature.

Contribution

It proves second moment estimates for primitive roots modulo composite numbers, a case previously unaddressed in the literature.

Findings

Established second moment bounds for primitive roots modulo composite n

Extended average results to the composite moduli case

Filled a gap in the understanding of primitive roots for composite moduli

Abstract

Artin's Conjecture on Primitive Roots states that a non-square nonunit integer $a$ is a primitive root modulo $p$ for the positive proportion of $p$. This conjecture remains open, but on average, there are many results due to P. J. Stephens. There is a natural generalization of the conjecture for composite moduli. We can consider $a$ as the primitive root modulo, $(\mathbb{Z}/n\mathbb{Z})^{*}$ if $a$ is an element of the maximal exponent in the group. The behavior is more complex for composite moduli, and the corresponding average results are provided by S. Li and C. Pomerance, and recently by the author. P. J. Stephens included the second moment results in his work, but for composite moduli, there were no such results previously. We prove that the corresponding second moment results in this case.