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Moments of real Dirichlet $L$-functions and multiple Dirichlet series

Martin Čech

Abstract

We consider the multiple Dirichlet series associated to the $k$th moment of real Dirichlet $L$-functions, and prove that it has a meromorphic continuation to a specific region in $\mathbb{C}^{k+1}$, which is conditional under the generalized Lindelöf hypothesis for $k\geq 5$. As a corollary, we obtain asymptotic formulas for the first three moments with a power-saving error term, and detect the 0- and 1-swap terms in related problems for any $k$ (conditionally under the Generalized Lindelöf Hypothesis), recovering the recent results of Conrey and Rodgers on long Dirichlet polynomials. The advantage of our method is its simplicity, since we don't need to modify the multiple Dirichlet series to obtain its meromorphic continuation. As a result, we obtain the asymptotic formulas directly in the form as they appear in the recipe predictions of Conrey, Farmer, Keating, Rubinstein and Snaith.

Moments of real Dirichlet $L$-functions and multiple Dirichlet series

Abstract

We consider the multiple Dirichlet series associated to the th moment of real Dirichlet -functions, and prove that it has a meromorphic continuation to a specific region in , which is conditional under the generalized Lindelöf hypothesis for . As a corollary, we obtain asymptotic formulas for the first three moments with a power-saving error term, and detect the 0- and 1-swap terms in related problems for any (conditionally under the Generalized Lindelöf Hypothesis), recovering the recent results of Conrey and Rodgers on long Dirichlet polynomials. The advantage of our method is its simplicity, since we don't need to modify the multiple Dirichlet series to obtain its meromorphic continuation. As a result, we obtain the asymptotic formulas directly in the form as they appear in the recipe predictions of Conrey, Farmer, Keating, Rubinstein and Snaith.
Paper Structure (12 sections, 15 theorems, 128 equations)

This paper contains 12 sections, 15 theorems, 128 equations.

Table of Contents

  1. Introduction and statement of results
  2. Preliminaries
  3. Dirichlet characters and $L$-functions
  4. Gauss sums
  5. Multivariable complex analysis
  6. Convex polyhedra
  7. Basic properties of $A(s_1,\dots,s_k,w)$
  8. Heuristic extra functional equation
  9. The functional equation in $w$ and meromorphic continuation
  10. Determining the region of meromorphic continuation
  11. Proof of Theorem \ref{['thm: main theorem']}
  12. Proof of the corollaries

Key Result

Theorem 1.3

Assume that $k\leq 4$ or that GLH holds. Then $A(s_1,\dots,s_k,w)$ has a meromorphic continuation to the region which is the intersection of the half-spaces (with $i_1,i_2,i_3,i_4$ pairwise different) and their reflections under the transformations $\sigma_J$, $J\subset\{1,\dots,k\}$. Its only poles in this region are at the points where $J\subset\{1,\dots,k\}$, with residues where $T(S\setminu

Theorems & Definitions (30)

  • Conjecture 1.1: Conrey, Farmer, Keating, Rubinstein, Snaith
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Proposition 2.1
  • Theorem 2.2: Bochner's Tube Theorem
  • Proposition 2.3
  • ...and 20 more