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

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 $δ$ such that if $x^{1-δ}\log^3 x = o(y)$, 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 $B=e^{-γ}\prod_p \left(1-\frac 1{(p-1)^2(p+1)}\right).$ This is an improvement over a statement in Kurlberg and Pomerance (see ~\cite{KP}): $\frac{1}{x^2} \sum_{a<x} \sum_{a<n<x} l_a(n) = \frac x{\log x} \exp \left(B \frac{\log\log x} {\log\log\log x} (1+o(1)) \right).$

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