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First moment of central values of some primitive Dirichlet $L$-functions with fixed order characters

Peng Gao, Liangyi Zhao

Abstract

We evaluate asymptotically the smoothed first moment of central values of families of primitive cubic, quartic and sextic Dirichlet $L$-functions, using the method of double Dirichlet series. Quantitative non-vanishing result for these $L$-values are also proved.

First moment of central values of some primitive Dirichlet $L$-functions with fixed order characters

Abstract

We evaluate asymptotically the smoothed first moment of central values of families of primitive cubic, quartic and sextic Dirichlet -functions, using the method of double Dirichlet series. Quantitative non-vanishing result for these -values are also proved.
Paper Structure (13 sections, 9 theorems, 76 equations)

This paper contains 13 sections, 9 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Residue symbols
  4. Gauss sums
  5. Estimating $L$-functions
  6. Analytic behavior of Dirichlet series associated with Gauss sums
  7. Some results on multivariable complex functions
  8. Proof of Theorem \ref{['firstmoment']}
  9. Initial Treatment
  10. First region of absolute convergence of $A_j(s,w)$
  11. Second region of absolute convergence of $A_j(s,w)$
  12. Residue of $A_j(s,w)$ at $s=1$
  13. Completion of the Proof

Key Result

Theorem 1.1

With the notation as above and assuming the truth of the generalized Lindelöf hypothesis, let $j=3, 4$ or $6$. We have, for $1/2>\Re(\alpha) \geq 0$, all large $Q$ and any $\varepsilon>0$, where $C_j$ is a positive constant explicitly given in eq:c and $\sum^*$ over $\chi$ means the sum runs over primitive characters $\chi$ such that $\chi^i, 1 \leq i \leq j-1$ remains primitive.

Theorems & Definitions (13)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.9
  • ...and 3 more