Cancellations between Kloosterman sums modulo a prime power with prime arguments

TL;DR

This paper establishes new bounds for cancellations in Kloosterman sums modulo prime powers with prime arguments over short intervals, using novel bilinear sum estimates that extend previous prime modulus results.

Contribution

It introduces a new technique for bounding Kloosterman sums modulo prime powers, enabling nontrivial estimates over much shorter ranges than prior methods.

Findings

Nontrivial bounds for Kloosterman sums modulo prime powers

Extension of cancellation results to shorter intervals

Development of a new bilinear sum estimate technique

Abstract

We obtain a nontrivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short interval, which in turn is based on a new estimate on bilinear sums of Kloosterman sums. These results are analogues of those obtained by various authors for Kloosterman sums modulo a prime. However the underlying technique is different and allows us to obtain nontrivial results starting from much shorter ranges.