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Moments and One level density of sextic Hecke $L$-functions of $\mathbb{Q}(ω)$

Peng Gao, Liangyi Zhao

Abstract

In this paper, we study moments of central values of sextic Hecke $L$-functions of $\mathbb{Q}(ω)$ and one level density result for the low-lying zeros of sextic Hecke $L$-functions of $\mathbb{Q}(ω)$. As a corollary, we deduce that, assuming GRH, at least $2/45$ of the members of the sextic family do not vanish at $s=1/2$.

Moments and One level density of sextic Hecke $L$-functions of $\mathbb{Q}(ω)$

Abstract

In this paper, we study moments of central values of sextic Hecke -functions of and one level density result for the low-lying zeros of sextic Hecke -functions of . As a corollary, we deduce that, assuming GRH, at least of the members of the sextic family do not vanish at .
Paper Structure (15 sections, 11 theorems, 101 equations)

This paper contains 15 sections, 11 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminaries
  4. Residue symbols and Kronecker symbol
  5. Gauss sums
  6. The approximate functional equation
  7. Analytic behavior of Dirichlet series associated with sextic Gauss sums
  8. The large sieve with sextic symbols
  9. Poisson Summation
  10. Proof of Theorem \ref{['firstmoment']}
  11. Evaluating $M_1$, the main term
  12. Estimating $M_1'$, the remainder term
  13. Estimating $M_2$
  14. Conclusion
  15. Proof of Theorem \ref{['sexticmainthm']}

Key Result

Theorem 1.1

Let $W: (0, \infty) \rightarrow \mathbb R$ be a smooth, compactly supported function. For $y \rightarrow \infty$ and any $\varepsilon > 0$, where $\chi_c=\left(\frac{\cdot}{c}\right)_6$ is the sextic residue symbol in $\mathbb Q(\omega)$ (defined in Section sec2.4) Moreover, $A$ is an explicit constant given in eq:c and $\sum^*$ denotes summation over squarefree elements of $\mathbb Z[\omega]$.

Theorems & Definitions (14)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.3
  • proof
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.9
  • ...and 4 more