Kloosterman sums with primes to composite moduli

TL;DR

This paper introduces a new estimate for Kloosterman sums involving primes and composite moduli, extending understanding of exponential sums and their applications to solving certain congruences.

Contribution

It provides a novel bound for Kloosterman sums with primes modulo composite numbers, applicable in a broader range than previous results.

Findings

New estimate for Kloosterman sums with primes and composite moduli

Applicable in the range q^{3/4+ε} ≤ X ≤ q^{3/2}

Used to prove solvability of specific congruences with inverse prime residues

Abstract

We obtain a new estimate for Kloosterman sum with primes $p\leqslant X$ to composite modulo $q$, that is, for the exponential sum of the type \[ \sum\limits_{p\leqslant X,\;p\,\nmid q}\exp{\biggl(\frac{2\pi i}{q}\bigl(a\overline{p}+bp\bigr)\,\biggr)},\quad (ab,q)=1,\quad p\overline{p}\equiv 1\pmod{q}, \] which is non-trivial in the case when $q^{\,3/4+\varepsilon}\leqslant X\ll q^{\,3/2}$. We also apply this estimate to the proof of solvability of some congruences with inverse prime residues $\pmod{q}$.