Short Kloosterman sums to powerful modulus

TL;DR

This paper provides an estimate for incomplete Kloosterman sums with powerful moduli, particularly for sums of length between exponential and square root of the modulus, advancing understanding in analytic number theory.

Contribution

It introduces new bounds for incomplete Kloosterman sums to powerful moduli within a specific length range, extending previous results in the field.

Findings

Established bounds for sums with length between exponential and square root of modulus

Extended the range of known estimates for Kloosterman sums to powerful moduli

Enhanced tools for analyzing exponential sums in number theory

Abstract

We obtain the estimate of incomplete Kloosterman sum to powerful modulus $q$. The length $N$ of the sum lies in the interval $e^{c(\log{q})^{2/3}}\le N\le \sqrt{q}$.