The average order of elements in the multiplicative group of a finite field (Q1950938)

For a positive integer \(n\), let \(\alpha(n)\) be the average multiplicative order of an invertible element modulo \(n\). The average value of \(\alpha(n)\) for \(n\) ranging in the interval \([1,x]\) was considered by \textit{J. von zur Gathen} et al. [J. Théor. Nombres Bordx. 16, No. 1, 107--123 (2004; Zbl 1079.11003)]. The reviewer [Ramanujan J. 9, No. 1--2, 33--44 (2005; Zbl 1155.11344)] considered the average values of \(\alpha(n)\) and \(\alpha(n)/n\) when \(n\) ranges over numbers of the form \(p-1\) with \(p\) a prime in \([1,x]\). In the paper under review, the authors look at the same problem when \(n\) ranges over numbers of the form \(p^k-1\). This is related to the average value of the multiplicative order in finite fields with \(q=p^k\) elements. Fixing \(k\), they show that the average value of \(\alpha(p^k-1)/(p^k-1)\) for \(p\) ranging over primes in \([1,x]\) is a positive constant \(K_k\), which they identify as an Euler product. Their result holds with some uniformity in \(k\) versus \(x\) (like \(k\leq \log x/(2\log\log x)\)). The proof uses the Bombieri-Vinogradov theorem on primes in arithmetic progressions.