On the Order of $a$ modulo $n$ on Average

TL;DR

This paper studies the average behavior of the multiplicative order of integers modulo n, providing an improved asymptotic formula under certain conditions, refining previous results by Kurlberg and Pomerance.

Contribution

It establishes a more precise asymptotic estimate for the average multiplicative order, improving upon prior bounds with new conditions and constants.

Findings

Derived an asymptotic formula for the average order of a modulo n.

Improved the error terms and conditions compared to previous work.

Identified the constant B involving an infinite product over primes.

Abstract

Let $a>1$ be an integer. Denote by $l_a(n)$ the multiplicative order of $a$ modulo integer $n\geq 1$. We prove that there is a positive constant $\delta$ such that if $x^{1-\delta}\log^3 x = o(y)$, then $$ \frac1y \sum_{a<y} \frac1x \sum_{\substack{{a<n<x}\\{(a,n)=1}}}l_a(n) = \frac x{\log x}\exp \left(B\frac{\log\log x}{\log\log\log x}(1+o(1))\right)$$ where $$ B=e^{-\gamma}\prod_p \left(1-\frac 1{(p-1)^2(p+1)}\right).$$ This is an improvement over a statement in Kurlberg and Pomerance (see ~\cite{KP}): $$\frac{1}{x^2} \sum_{a<x} \sum_{a<n<x} l_a(n) = \frac x{\log x} \exp \left(B \frac{\log\log x} {\log\log\log x} (1+o(1)) \right).$$