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Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues

Jiseong Kim

Abstract

By assuming Vinogradov-Korobov type zero-free regions and the generalized Ramanujan-Petersson conjecture, we establish nontrivial upper bounds for almost all short sums of Fourier coefficients of Hecke-Maass cusp forms for $SL(n,\mathbb{Z})$. As applications, we obtain nontrivial upper bounds for the averages of shifted sums involving coefficients of the Hecke-Maass cusp forms for $SL(n,\mathbb{Z})$. Furthermore, we present a conditional result regarding sign changes of these coefficients.

Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues

Abstract

By assuming Vinogradov-Korobov type zero-free regions and the generalized Ramanujan-Petersson conjecture, we establish nontrivial upper bounds for almost all short sums of Fourier coefficients of Hecke-Maass cusp forms for . As applications, we obtain nontrivial upper bounds for the averages of shifted sums involving coefficients of the Hecke-Maass cusp forms for . Furthermore, we present a conditional result regarding sign changes of these coefficients.

Paper Structure

This paper contains 14 sections, 17 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Holomorphic Hecke cusp forms
  3. Hecke-Maass cusp forms for $\bf{SL(n, \mathbb{Z})}$
  4. The generalized Ramanujan-Petersson conjecture
  5. Zero-free regions and shifted convolution sum
  6. Main results
  7. Lemmas
  8. Proof of Theorems and corollaries
  9. Proof of theorem \ref{['Theo1']}
  10. Proof of Theorem \ref{['Theo1']}
  11. Proof of Theorem \ref{['Theorem 3.10']}
  12. Proof of Corollary \ref{['Cor2']}
  13. Proof of Corollary \ref{['Cor16']}
  14. Acknowledgements

Key Result

Theorem 1.1

Assume the generalized Ramanujan-Petersson conjecture and the $SL(n,\mathbb{Z})$ Vinogradov-Korobov zero-free regions. Let $X$ be sufficiently large, and let $e^{ (\log X)^{1-\epsilon}} \ll_{\epsilon} h_{1} \leq h_{2} \ll_{\epsilon} X^{1-\epsilon}$ where $\epsilon>0$ is fixed small. Then

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • proof
  • Lemma 1.3: MJM
  • Remark 1.4
  • Lemma 1.5
  • proof
  • Remark 1.6
  • Corollary 1.7
  • Corollary 1.8
  • ...and 29 more