title: On the generalization of Lehmer problem and high-dimension Kloosterman sums.
abstract: For any fixed integer k ≥ 2 and integer r with (r, p) = 1, it is clear that there exist k integers 1 ≤ a i ≤ p - 1(i = 1, 2, …, k) such that a 1 a 2 ⋯ a k ≡ r mod p. Let N(k, r; p) denote the number of all (a 1, a 2, ⋯ a k ) such that a 1 a 2 ⋯ a k ≡ r mod p and 2†(a 1 + a 2 + ⋯ + a k ). In this paper, we will use the analytic method and the estimate for high-dimension Kloosterman sums to study the asymptotic properties of N(k, r; p) and give two interesting asymptotic formulae for it.
authors.affiliation: College of Mathematics and Statistics, Hainan Normal University, Hainan 571158, China.
authors.forename: Guohui
authors.initials: G
authors.lastname: Chen
authors.affiliation: School of Mathematics, Northwest University, Xi'an, Shaanxi 710127, China.
authors.forename: Han
authors.initials: H
authors.lastname: Zhang
mesh_terms: D055641:Mathematical Concepts