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Wide moments of $L$-functions II: Dirichlet $L$-functions

Asbjørn Christian Nordentoft

Abstract

We study wide moments of Dirichlet $L$-functions using analytic properties of the Lerch zeta function. Among other things we obtain an asymptotic expansion of wide moments of Dirichlet $L$-functions (with arbitrary twists) extending results of Heath-Brown. We also give applications to non-vanishing.

Wide moments of $L$-functions II: Dirichlet $L$-functions

Abstract

We study wide moments of Dirichlet -functions using analytic properties of the Lerch zeta function. Among other things we obtain an asymptotic expansion of wide moments of Dirichlet -functions (with arbitrary twists) extending results of Heath-Brown. We also give applications to non-vanishing.

Paper Structure

This paper contains 15 sections, 17 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. $L$-functions and automorphic periods
  3. Wide moments
  4. Statement of results
  5. Analytic properties of Lerch zeta functions
  6. Lerch zeta functions as Fourier transforms of Dirichlet $L$-functions
  7. Twisted version of the Lerch zeta function
  8. Moments of Lerch zeta functions
  9. Moments of the Hurwitz zeta function
  10. Moments of the twisted periodic zeta function
  11. Double moments of the Lerch zeta function
  12. Applications to wide moments of Dirichlet $L$-functions and non-vanishing
  13. Wide moments
  14. Further examples of wide moments
  15. Non-vanishing

Key Result

Theorem 1.1

For $i=1,\ldots, m$ with $m\geq 2$, let $\psi_i\text{ \rm mod } \ell_i$ be Dirichlet characters. Then we have where $T_{\overline{d},\psi_i}$ depends only on the residue class $\overline{d}$ of $d$ modulo $\ell:=\mathrm{lcm}(\ell_1,\ldots, \ell_m)$ and on the Dirichlet characters $\psi_i$. Furthermore, given $a\text{ \rm mod } \ell$, we have the following asymptotic expansion for any $K\geq 1$:

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 28 more