Uniform bounds for Kloosterman sums of half-integral weight, same-sign case

TL;DR

This paper extends uniform bounds for half-integral weight Kloosterman sums to the same-sign case, enhancing previous bounds and enabling broader applications in number theory.

Contribution

It proves a uniform bound for sums of half-integral weight Kloosterman sums in the same-sign case, generalizing previous results to new parameter ranges.

Findings

Established uniform bounds for the same-sign case of half-integral weight Kloosterman sums.

Extended previous bounds to include the case when  m  n > 0.

Facilitated new applications in partition formulas and number theory.

Abstract

In the previous paper [Sun23], the author proved a uniform bound for sums of half-integral weight Kloosterman sums. This bound was applied to prove an exact formula for partitions of rank modulo 3. That uniform estimate provides a more precise bound for a certain class of multipliers compared to the 1983 result by Goldfeld and Sarnak and generalizes the 2009 result from Sarnak and Tsimerman to the half-integral weight case. However, the author only considered the case when the parameters satisfied $\tilde m\tilde n<0$. In this paper, we prove the same uniform bound when $\tilde m\tilde n>0$ for further applications.