A Note on the Mean Value of Numbers of the Solutions of x^α ≡ 1(mod n)

Abstract

Let T = T(p, q, α) be the number of solutions of the congruence x^α ≡ 1 (mod p^ηq^θ). Let A and B be sets of primes satisfying x₁ < p ≤ x₂ and y₁ < q ≤ y₂, respectively. A mean value estimation of $\frac{1} {{\left| {A\left\| {B\left| {} \right.} \right.} \right.}}{\sum\nolimits_{p \in A} {{\sum\nolimits_{q \in B} {\log \;T{\left( {p,q,\alpha } \right)}} }} }$is given.