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

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}): $$\frac{1}{x^2} \sum_{a<x} \sum_{a<n<x} l_a(n) = \frac x{\log x} \exp \left(B \frac{\log\log x} {\log\log\log x} (1+o(1)) \right).$$