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

TL;DR

This paper develops a new estimate for Kloosterman sums over primes and applies it to determine the solvability of a specific congruence involving inverse residues in prime variables, providing an asymptotic count of solutions.

Contribution

It introduces a novel estimate for Kloosterman sums over primes and applies it to solve a congruence problem with prime variables, extending previous results.

Findings

Established a new estimate for Kloosterman sums over primes

Derived an asymptotic formula for the number of solutions to the congruence

Proved solvability conditions for the congruence when q is coprime to 6

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 $p_{1},\ldots, p_{k}\le N$, $N\le q^{1-\delta}$. Here $g(x)\,\equiv\,a\overline{x}+bx\pmod{q}$, 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.