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First moment of central values of Hecke $L$-functions with Fixed Order Characters

Peng Gao, Liangyi Zhao

TL;DR

The paper develops a unified framework to study the smoothed first moment of central values for families of Hecke L-functions attached to fixed-order characters over imaginary quadratic fields of class number one, using double Dirichlet series and Gauss-sum–driven functional equations. It provides asymptotic formulas for quadratic, cubic, quartic, and sextic families, with a GRH-based $O(X^{1/4+\varepsilon})$ error term in the quadratic case and unconditional results for higher orders. The method replaces the need for extensive families of functional equations with a general functional equation for nonprimitive quadratic Hecke L-functions and a Soundararajan–Young style analysis of Gauss sums, enabling meromorphic continuation to a large region and explicit residue computations that yield the main terms. These results extend the understanding of mean values of central L-values to Hecke L-functions over imaginary quadratic fields and give sharper error terms, highlighting the power of the double Dirichlet series approach in this arithmetic setting.

Abstract

We evaluate asymptotically the smoothed first moment of central values of families of quadratic, cubic, quartic and sextic Hecke $L$-functions over various imaginary quadratic number fields of class number one, using the method of double Dirichlet series. In particular, we obtain asymptotic formulas for the quadratic families with error terms of size $O(X^{1/4+\varepsilon})$ under the generalized Riemann hypothesis.

First moment of central values of Hecke $L$-functions with Fixed Order Characters

TL;DR

The paper develops a unified framework to study the smoothed first moment of central values for families of Hecke L-functions attached to fixed-order characters over imaginary quadratic fields of class number one, using double Dirichlet series and Gauss-sum–driven functional equations. It provides asymptotic formulas for quadratic, cubic, quartic, and sextic families, with a GRH-based error term in the quadratic case and unconditional results for higher orders. The method replaces the need for extensive families of functional equations with a general functional equation for nonprimitive quadratic Hecke L-functions and a Soundararajan–Young style analysis of Gauss sums, enabling meromorphic continuation to a large region and explicit residue computations that yield the main terms. These results extend the understanding of mean values of central L-values to Hecke L-functions over imaginary quadratic fields and give sharper error terms, highlighting the power of the double Dirichlet series approach in this arithmetic setting.

Abstract

We evaluate asymptotically the smoothed first moment of central values of families of quadratic, cubic, quartic and sextic Hecke -functions over various imaginary quadratic number fields of class number one, using the method of double Dirichlet series. In particular, we obtain asymptotic formulas for the quadratic families with error terms of size under the generalized Riemann hypothesis.
Paper Structure (19 sections, 14 theorems, 147 equations)

This paper contains 19 sections, 14 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Imaginary quadratic number fields
  4. Residue symbols
  5. Gauss sums
  6. Hecke $L$-functions
  7. Functional Equation of Hecke $L$-functions
  8. Bounding $L$-functions
  9. Some results on multivariable complex functions
  10. Proof of Theorem \ref{['Thmfirstmoment']} and \ref{['Thmfirstmomentjlarge']}
  11. Initial Treatment
  12. First region of absolute convergence of $A_j(s,w)$
  13. Residue of $A_j(s,w)$ at $s=1$
  14. Second region of absolute convergence of $A_j(s,w)$
  15. The case $j > 2$
  16. ...and 4 more sections

Key Result

Theorem 1.1

With the notation as above and assuming the truth of GRH, for any imaginary quadratic number field $K$, we have for $1/2>\Re(\alpha)>0$, all large $X$ and any $\varepsilon>0$,

Theorems & Definitions (19)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • Lemma 2.10
  • ...and 9 more