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First Moment of Quadratic Hecke $L$-Functions with Lower Order Term

Peng Gao, Liangyi Zhao

TL;DR

This paper analyzes the first moment of the family of primitive quadratic Hecke L-functions in the Gaussian field $K=\mathbb{Q}(i)$ using the double Dirichlet series method. Under RH and the Lindelöf hypothesis, it proves meromorphic continuation and identifies explicit residue terms that generate a main term with secondary components in the asymptotics of the smoothed first moment, with an error term that is $O(X^{\varepsilon})$ smaller than the main contribution. The authors derive a central-value formula with a sharp $X^{1/4+\varepsilon}$-type error under GRH and show a distinct behavior for square-free conductors versus all conductors via a contour-shifting argument and careful residue calculations. The approach connects function-field intuition (as seen in Florea) with number-field phenomena (via Gaussian fields) and highlights the power of double Dirichlet series in isolating main and secondary terms for families of $L$-functions.

Abstract

We evaluate the first moment of the family of primitive quadratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series under the Riemann hypothesis and the Lindelöf hypothesis. We obtain asymptotic formulas with secondary main terms and error terms of size that is one quarter of that of the main term.

First Moment of Quadratic Hecke $L$-Functions with Lower Order Term

TL;DR

This paper analyzes the first moment of the family of primitive quadratic Hecke L-functions in the Gaussian field using the double Dirichlet series method. Under RH and the Lindelöf hypothesis, it proves meromorphic continuation and identifies explicit residue terms that generate a main term with secondary components in the asymptotics of the smoothed first moment, with an error term that is smaller than the main contribution. The authors derive a central-value formula with a sharp -type error under GRH and show a distinct behavior for square-free conductors versus all conductors via a contour-shifting argument and careful residue calculations. The approach connects function-field intuition (as seen in Florea) with number-field phenomena (via Gaussian fields) and highlights the power of double Dirichlet series in isolating main and secondary terms for families of -functions.

Abstract

We evaluate the first moment of the family of primitive quadratic Hecke -functions in the Gaussian field using the method of double Dirichlet series under the Riemann hypothesis and the Lindelöf hypothesis. We obtain asymptotic formulas with secondary main terms and error terms of size that is one quarter of that of the main term.
Paper Structure (11 sections, 15 theorems, 88 equations)

This paper contains 11 sections, 15 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quadratic residue symbol and quadratic Hecke character
  4. Functional equations for quadratic Hecke $L$-functions
  5. A mean value estimate for quadratic Hecke $L$-functions
  6. Multivariable complex analysis
  7. Proof of Theorem \ref{['thm:MDS']}
  8. First region of convergence
  9. Second region of convergence
  10. Completion of the proof
  11. Proof of Theorem \ref{['Theorem for all characters']}

Key Result

Theorem 1.1

With the notation as above and the Lindelöf hypothesis, let $\Phi(t)$ be a non-negative smooth function that is compactly supported in $[1/2, 2]$ with Mellin transform $\widehat{\Phi}(s)$. Set Fix $s\in\mathbb C$ with $1/3<\Re(s)<1$, $s\neq 1/2$. Then for any $\varepsilon>0,$ where and where $R_{K,1}(s), R_{K,2}(s)$ are explicitly defined in Theorem thm:MDS.

Theorems & Definitions (24)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 14 more