Kloosterman sums with primes and solvability of one congruence with inverse residues. I

Abstract

In the paper, we establish a new estimate for Kloosterman sum over primes with respect to an arbitrary modulus $q$. This estimate together with some recent results of the second author are applied to the problem of solvability of the congruence \[ g(p_{1})\, + \,\ldots\, + \,g(p_{k}) \,\equiv\, m\pmod{q} \] in prime variables $p_1,\dots ,p_k\le N$, $N\le q^{1-\delta }$. Here $g(x)\equiv a\overline{x}+bx(\mathrm{mod}q)$, where $a,b,m$ and $k\ge 3$ are arbitrary integers, $(ab,q)=1$. The main result of the paper gives an asymptotic formula for the number of solutions for the case when $q$ is coprime to $6$. In this version, we correct some typos and small errors.

37 pages

References in corpus (1)

• Kloosterman sums with primes to composite moduli