On a problem of Arnold: The average multiplicative order of a given integer

Abstract

For coprime integers $g$ and $n$, let ${\ell }_g\left(n\right)$ denote the multiplicative order of $g$ modulo $n$. Motivated by a conjecture of Arnold, we study the average of ${\ell }_g\left(n\right)$ as $n\le x$ ranges over integers coprime to $g$, and $x$ tending to infinity. Assuming the generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptotics of the average of ${\ell }_g\left(p\right)$ as $p\le x$ ranges over primes.