On the generalization of the D. H. Lehmer problem
Let \(n \geq 2\) be a fixed positive integer, \(q \geq 3\) and \(c\) be two integers with \((n, q) = (c, q) = 1\). Let \(r_n (\delta _{1}, \delta _{2}, c; q) (0 < \delta _{1}, \delta _{2} \leq 1)\) denote the number of all pairs of integers \(a, b\) satisfying \(ab \equiv c(\text{mod} q), 1 \leq a \leq \delta _{1} q, 1 \leq b \leq \delta _{2} q, (a, q) = (b, q) = 1\) and \(n\nmid (a + b)\). In this paper paper the authors study the asymptotic properties of \(r_n (\delta _{1}, \delta _{2}, c; q)\), and give a sharp asymptotic formula for it. Theorem: It holds that \[ r_n (\delta _1, \delta _2, c; q)=(1-\frac 1n)\delta_1\delta_2\varphi(q) + O(q^{\frac 12} d^6(q) \log^2 q), \] where \(\varphi(n)\) is the Euler function, \(d(n)\) is the divisor function, and the \(O\) constant only depends on \(n\). In the special cases \(\delta _1= \delta _2=c=1\) and \(n=2\) and \(\delta _1= \frac Nq, \delta _2=c=1\) and \(n=2\) they obtain results similar to those of \textit{Wenpeng Zhang} [Compos. Math. 86, No. 3, 307--316 (1993; Zbl 0783.11002) and Compos. Math. 91, No. 1, 47--56 (1994; Zbl 0798.11001)].
scientific article; zbMATH DE number 5601621