On the Generalization of Lehmer Problem and High-Dimension Kloosterman Sums

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.