Type: Article

An estimate for Kloosterman sums with primes and its application

M. Z. Garaev;

Abstract:

Suppose that p is a large prime. In this paper, we prove that, for any natural number N < p the following estimate holds: max((a, p)=1) vertical bar Sigma(q <= N) e(2 pi iaq*/p)vertical bar <= N(15/16) + N(2/3)p(1/4))p(o(1)), where q is a prime and q* is the least natural number satisfying the congruence qq* = 1 (modp). This estimate implies the following statement: if p > N > p(16/17+epsilon), where epsilon > 0, and if we have lambda not equivalent to 0 (modp), then the number J of solutions of the congruence q(1)(q(2) + q(3)) lambda (mod p) for the primes q(1), q(2), q(3) <= N can be expressed as J = pi(N)(3)/p (1 + O(p(-delta))), delta = delta(epsilon) > 0. This statement improves a recent result of Friedlander, Kurlberg, and Shparlinski in which the condition p > N > p(38/39+epsilon) was required.