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Uniform bounds for sums of Kloosterman sums of half integral weight

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Abstract

For \(m,n>0\) or \(mn<0\) we estimate the sums

$$\begin{aligned} \sum _{c \le x} \frac{S(m,n,c,\chi )}{c}, \end{aligned}$$

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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Acknowlegements

The author thanks Professor Scott Ahlgren for his careful reading of the manuscript and both of the referees for their thorough reports. The author is grateful to Nick Andersen for his insightful comments.

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  1. Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL, 61801, USA

    Alexander Dunn

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Dunn, A. Uniform bounds for sums of Kloosterman sums of half integral weight. Res. number theory 4, 45 (2018). https://doi.org/10.1007/s40993-018-0138-6

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