On the order of \(a\) modulo \(n\), on average (Q2833079)

Let \(n\geq 1\) and \(a\) are coprime integers, and denote by \(\mathrm{ord}_n(a)\) the multiplicative order of \(a\) modulo \(n\), which is the smallest positive integer \(d\) satisfying \(a^d\equiv 1\pmod{n}\). In the paper under review, the author shows that there exists \(\delta>0\) such that if \(x^{1-\delta}=o(y)\), then NEWLINE\[NEWLINE \frac{1}{y}\sum_{a<y}\frac{1}{x}\sum_{\substack{ a<n<x\\(a,n)=1}} \mathrm{ord}_n(a)=\frac{x}{\log x}\exp\left(B\frac{\log\log x}{\log\log\log x}(1+o(1))\right), NEWLINE\]NEWLINE where NEWLINE\[NEWLINE B=e^{-\gamma}\prod_{p}\left(1-\frac{1}{(p-1)^2(p+1)}\right). NEWLINE\]