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Ratios conjecture of quartic $L$-functions of prime moduli

Peng Gao, Liangyi Zhao

Abstract

We apply the method of multiple Dirichlet series to develop $L$-functions ratios conjecture with one shift in both the numerator and denominator in certain ranges for the family of quartic Hecke $L$-functions of prime moduli over the Gaussian field under the generalized Riemann hypothesis. As consequences, we evaluate asymptotically the first moment of central values as well as the one-level density of the same family of $L$-functions.

Ratios conjecture of quartic $L$-functions of prime moduli

Abstract

We apply the method of multiple Dirichlet series to develop -functions ratios conjecture with one shift in both the numerator and denominator in certain ranges for the family of quartic Hecke -functions of prime moduli over the Gaussian field under the generalized Riemann hypothesis. As consequences, we evaluate asymptotically the first moment of central values as well as the one-level density of the same family of -functions.
Paper Structure (22 sections, 27 theorems, 208 equations)

This paper contains 22 sections, 27 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quartic residue symbols and Gauss sums
  4. Functional equations for Hecke $L$-functions
  5. Estimations on $L$-functions
  6. The large sieve with quadratic symbols
  7. Some results on multivariable complex functions
  8. Metaplectic Eisenstein series
  9. The Kubota symbol
  10. Eisenstein series
  11. Singular values of Eisenstein series
  12. Relations between residues
  13. Proof of Proposition \ref{['prop: Fbound']}
  14. Analytic behavior of Dirichlet series associated with quartic Gauss sums
  15. Completion of the proof of Proposition \ref{['prop: Fbound']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

With the notation as above and assuming the truth of GRH, let $X$ be a large real number, $w(t)$ a non-negative Schwartz function with $\widehat{w}(s)$ being its Mellin transform. Set We also set $\delta(\alpha)=2/11$ when $\Re(\alpha) <0$ and $\delta(\alpha)=0$ otherwise. Then for $\Re(\alpha) > -1/11$ and $\Re(\beta)>0$ such that $E(\alpha,\beta)<1$, where the "$*$" on the sum over $\varpi$ me

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 23 more