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

Abstract

In the previous paper [Sun24], 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.