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

Qihang Sun

TL;DR

The paper delivers a uniform bound for sums of half-integral weight Kloosterman sums in the same-sign regime $\tilde{m}\tilde{n}>0$, extending prior work that treated $\tilde{m}\tilde{n}<0$ and broadening applications to partitions and half-integral weight automorphic forms. It employs a half-integral weight trace formula with admissible multipliers, decomposes the spectral side into holomorphic, Maass cusp form, and Eisenstein series contributions, and develops precise bounds for the transformed test functions $\widetilde{\phi}$ and $\widehat{\phi}$, including the handling of exceptional spectrum. A specially crafted test function isolates exceptional eigenvalues, enabling a rigorous proof that combines level-lifting, Voronoi-type spectral bounds, and careful dyadic summation across spectral ranges. The results reinforce a half-integral weight analogue of the Linnik–Selberg framework and have implications for harmonic Maass forms and rank-partition theory, providing sharper uniform estimates and extending the range of weight and multiplier systems for which uniform bounds hold.

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.

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

TL;DR

The paper delivers a uniform bound for sums of half-integral weight Kloosterman sums in the same-sign regime , extending prior work that treated and broadening applications to partitions and half-integral weight automorphic forms. It employs a half-integral weight trace formula with admissible multipliers, decomposes the spectral side into holomorphic, Maass cusp form, and Eisenstein series contributions, and develops precise bounds for the transformed test functions and , including the handling of exceptional spectrum. A specially crafted test function isolates exceptional eigenvalues, enabling a rigorous proof that combines level-lifting, Voronoi-type spectral bounds, and careful dyadic summation across spectral ranges. The results reinforce a half-integral weight analogue of the Linnik–Selberg framework and have implications for harmonic Maass forms and rank-partition theory, providing sharper uniform estimates and extending the range of weight and multiplier systems for which uniform bounds hold.

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 . In this paper, we prove the same uniform bound when for further applications.
Paper Structure (15 sections, 20 theorems, 182 equations)

This paper contains 15 sections, 20 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Application
  3. Background
  4. Kloosterman sums
  5. Maass forms
  6. Holomorphic cusp forms of half-integral weight
  7. Trace formula
  8. Properties of admissible multipliers
  9. Bounds on $\widetilde{\phi}$ and $\widehat{\phi}$
  10. A special test function
  11. Proof of Theorem \ref{['mainThm']} and Theorem \ref{['mainThmLastSec']}
  12. On the case $k=\frac{1}{2}$
  13. Contribution from holomorphic forms
  14. Contribution from Maass cusp forms and Eisenstein series.
  15. On the case $k=-\frac{1}{2}$

Key Result

Lemma 1.2

Let $\nu=(\frac{|D|}{\cdot})\nu_\theta$ or $\nu=(\frac{|D|}{\cdot})\nu_\eta$ where $D$ is a fundamental discriminant and $\nu_\theta$ and $\nu_\eta$ are the multiplier system for the standard theta function and Dedekind's eta function, respectively (see thetaMultiplier and etaMultiplier). Then both

Theorems & Definitions (35)

  • Definition 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Remark
  • Conjecture 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 3.3: Proskurin2005
  • Remark
  • Proposition 3.4
  • ...and 25 more