On the density of some sets of integers (Q2641345)

Let a and n be given integers with \(| a| \geq 2\) and \(n\geq 2\), and denote by N(x,n,a) the number of positive integers \(m\leq x\) coprime to a for which the order of a modulo m (i.e., the least positive integer r with \(a^ r\equiv 1\) modulo m) is coprime to n. \textit{R. W. K. Odoni} [J. Number Theory 13, 303-319 (1981; Zbl 0471.10031)] showed that, as \(x\to \infty\), \(N(x,n,a)\sim \beta x(\log x)^{\alpha -1},\) where \(\beta\) and \(\alpha\) are positive constants depending on a and n. In an earlier paper [Acta Arith. 43, 177-190 (1984; Zbl 0531.10049)], the author gave an estimate with logarithmic error term for the corresponding counting function when m is restricted to prime moduli. Using this estimate and a general result of \textit{B. V. Levin} and \textit{A. S. Fainleib} [Usp. Mat. Nauk 22, No.3(135), 119-197 (1967; Zbl 0204.065), Engl. translation in Russ. Math. Surveys 22, No.3, 119-204 (1967)] he shows in the present paper that, for any fixed \(\epsilon >0\), \[ N(x,n,a)=\beta \frac{x}{(\log x)^{1-\alpha}}+O(\frac{x}{(\log x)^{2-\alpha -\epsilon}}). \]