Estimation of Kloosterman Sums with Primes and Its Application

M. Z. Garaev

Matematičeskie zametki, Tome 88 (2010) no. 3, pp. 365-373

Résumé

Suppose that $p$ is a large prime. In this paper, we prove that, for any natural number $N$ the following estimate holds: $$ \max_{(a,p)=1}\biggl|\sum_{q\le N}e^{2\pi iaq^*/p}\biggr|\le(N^{15/16}+N^{2/3}p^{1/4})p^{o(1)}, $$ where $q$ is a prime and $q^*$ is the least natural number satisfying the congruence $qq^*\equiv1\,(\operatorname{mod}p)$. This estimate implies the following statement: if $p>N>p^{16/17+\varepsilon}$, where $\varepsilon>0$, and if we have $\lambda\not\equiv0\,(\operatorname{mod}p)$, then the number $J$ of solutions of the congruence $$ q_1(q_2+q_3)\equiv\lambda\quad(\operatorname{mod}p) $$ for the primes $q_1,q_2,q_3\le N$ can be expressed as $$ J=\frac{\pi(N)^3}p(1+O(p^{-\delta})),\qquad \delta=\delta(\varepsilon)>0. $$ This statement improves a recent result of Friedlander, Kurlberg, and Shparlinski in which the condition $p>N>p^{38/39+\varepsilon}$ was required.

Détail

Keywords: Kloosterman sum, Cauchy–Bunyakovskii inequality, Dirichlet's principle, Vinogradov sieve, Dirichlet $L$-function, trigonometric sum, Manholdt function.