The Average Order of a Permutation

Abstract

We show that the average order $\mu_n$ of a permutation in $S_n$ satisfies $$ \log\mu_n\ =\ C\sqrt{\frac n{\log n}} + O\left(\frac{\sqrt n\log\log n}{\log n}\right), $$ which refines earlier results of Erdős and Turán, Schmutz, and Goh and Schmutz.