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) equivalent to 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) equivalent to r mod p and 2 dagger (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.