Uniform bounds for sums of Kloosterman sums of half integral weight

Alexander Dunn

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arXiv · 1708.03003

Uniform bounds for sums of Kloosterman sums of half integral weight

Alexander Dunn

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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.

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Alexander Dunn. 2017-08-09. Uniform bounds for sums of Kloosterman sums of half integral weight. https://arxiv.org/abs/1708.03003

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