Uniform bounds for sums of Kloosterman sums of half integral weight

TL;DR

This paper establishes uniform bounds for sums of half-integral weight Kloosterman sums, extending previous results and providing stronger estimates in certain cases, with implications for analytic number theory.

Contribution

It introduces new uniform bounds for sums of half-integral weight Kloosterman sums, improving upon prior estimates and incorporating refinements based on factorization and Ramanujan-Petersson conjecture.

Findings

Uniform bounds for sums of Kloosterman sums with half-integral weight.

Stronger estimates in the case $mn<0$ compared to previous work.

Refined bounds depending on factorization and Ramanujan-Petersson exponent.

Abstract

For $m,n>0$ and $mn<0$ we estimate the sums \begin{equation*} \sum_{c \leq x} \frac{S(m,n,c,\chi)}{c}, \end{equation*} where the $S(m,n,c,\chi)$ are Kloosterman sums attached to a multiplier $\chi$ of weight $1/2$ on the full modular group. Our estimates are uniform in $m, n$ and $x$ in analogy with the bounds for the case $mn<0$ due to Ahlgren-Andersen, and those of Sarnak-Tsimerman for the trivial multiplier when $m,n>0$. In the case $mn<0$, our estimates are stronger in the $mn$-aspect than those of Ahlgren-Andersen. We also obtain a refinement whose quality depends on the factorization of $24m-23$ and $24n-23$ as well as the best known exponent for the Ramanujan-Petersson conjecture.