Estimation of Kloosterman Sums with Primes and Its Application

摘要

Suppose that p is a large prime. In this paper, we prove that, for any natural number N < p the following estimate holds: max_((z,p)=1)∣∑_(q≤N)e~(2πiaq*/p)∣≤(N~(15/16))+N~(2/3)p~(1/4)p~(°(1)) where q is a prime and q~* is the least natural number satisfying the congruence qq~* = 1 (mod p). This estimate implies the following statement: if p > N > p~(16)/~(17+∈) where e > 0, and if we have λ≠0 (modp), then the number J of solutions of the congruence q_1(q_2 + q_3) = λ (mod p) for the primes q_1, q_2, q_z ≤ N can be expressed as This statement improves a recent result of Friedlander, Kurlberg, and Shparlinski in which the condition p > N > p~(38/39+∈) was required.