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Multiple Dirichlet series predictions for moments of $L$-functions: unitary, symplectic and orthogonal examples

Siegfred Baluyot, Martin Čech

TL;DR

This work develops a multiple Dirichlet series (MDS) framework to predict shifted moments for four families of $L$-functions and systematically compares it to the Conrey–Farmer–Keating–Rubinstein–Snaith (CFKRS) recipe. For the unitary, symplectic, and orthogonal families, the authors demonstrate a precise one-to-one correspondence between residues of the MDS and the main terms in the CFKRS predictions, and they show that the MDS approach recovers the full CFKRS predictions with power-saving errors. In the elliptic-curve twist family, a root-number–coefficient correlation necessitates a modified version of the CFKRS recipe to align with the MDS predictions. The appendix confirms Shen’s first-moment result within this unified framework, and the paper thereby emphasizes that residues from MDS encapsulate the same information as CFKRS terms across these symmetry types, providing a unifying analytic perspective on moments of $L$-functions.

Abstract

We devise heuristics using multiple Dirichlet series to predict asymptotic formulas for shifted moments of (1) the family of Dirichlet $L$-functions of all even primitive characters of conductor $\leq Q$, with $Q$ a parameter tending to infinity, (2) the family of quadratic Dirichlet $L$-functions, (3) the family of quadratic twists of an $L$-function associated to a fixed Hecke eigencuspform for the full modular group, and (4) the family of quadratic twists of an $L$-function of a fixed arbitrary elliptic curve over $\mathbb{Q}$ that has a non-square conductor. For each of these families, the resulting predictions agree with the predictions of the recipe developed by Conrey, Farmer, Keating, Rubinstein, and Snaith, except for (4), where the recipe requires a slight modification due to a correlation between the Dirichlet coefficients and the root number of the corresponding $L$-functions. We find a one-to-one correspondence between the residues from the multiple Dirichlet series analysis and the terms from the recipe prediction.

Multiple Dirichlet series predictions for moments of $L$-functions: unitary, symplectic and orthogonal examples

TL;DR

This work develops a multiple Dirichlet series (MDS) framework to predict shifted moments for four families of -functions and systematically compares it to the Conrey–Farmer–Keating–Rubinstein–Snaith (CFKRS) recipe. For the unitary, symplectic, and orthogonal families, the authors demonstrate a precise one-to-one correspondence between residues of the MDS and the main terms in the CFKRS predictions, and they show that the MDS approach recovers the full CFKRS predictions with power-saving errors. In the elliptic-curve twist family, a root-number–coefficient correlation necessitates a modified version of the CFKRS recipe to align with the MDS predictions. The appendix confirms Shen’s first-moment result within this unified framework, and the paper thereby emphasizes that residues from MDS encapsulate the same information as CFKRS terms across these symmetry types, providing a unifying analytic perspective on moments of -functions.

Abstract

We devise heuristics using multiple Dirichlet series to predict asymptotic formulas for shifted moments of (1) the family of Dirichlet -functions of all even primitive characters of conductor , with a parameter tending to infinity, (2) the family of quadratic Dirichlet -functions, (3) the family of quadratic twists of an -function associated to a fixed Hecke eigencuspform for the full modular group, and (4) the family of quadratic twists of an -function of a fixed arbitrary elliptic curve over that has a non-square conductor. For each of these families, the resulting predictions agree with the predictions of the recipe developed by Conrey, Farmer, Keating, Rubinstein, and Snaith, except for (4), where the recipe requires a slight modification due to a correlation between the Dirichlet coefficients and the root number of the corresponding -functions. We find a one-to-one correspondence between the residues from the multiple Dirichlet series analysis and the terms from the recipe prediction.
Paper Structure (17 sections, 8 theorems, 118 equations)

This paper contains 17 sections, 8 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Unitary family: Dirichlet $L$-functions of even primitive characters of conductor $\leq Q$
  4. Unitary family: carrying out the CFKRS recipe
  5. Unitary family: predictions via multiple Dirichlet series
  6. Symplectic family: Dirichlet $L$-functions of real primitive characters
  7. Symplectic family: carrying out the CFKRS recipe
  8. Symplectic family: predictions via multiple Dirichlet series
  9. Orthogonal family: quadratic twists of modular Hecke eigenforms
  10. Orthogonal family: carrying out the CFKRS recipe
  11. Orthogonal family: predictions via multiple Dirichlet series
  12. Quadratic twists of elliptic curves
  13. Quadratic twists of elliptic curves: carrying out the CFKRS recipe
  14. Quadratic twists of elliptic curves: predictions via multiple Dirichlet series
  15. Appendix: the first moment of quadratic twists of elliptic curves
  16. ...and 2 more sections

Key Result

Proposition 2.4

The multiple Dirichlet series eqn: Adef is absolutely convergent if $\mathrm{Re}(w)>2$ and $\mathrm{Re}(s_j)>1$ and $\mathrm{Re}(z_j)>1$ for all $j$. It has a meromorphic continuation to the region $\mathrm{Re}(w)>1$, $\mathrm{Re}(s_j)>2,\ \mathrm{Re}(z_j)>2$, where it has a simple pole at $w=2$ wit where $B_{M,N}(S,Z)$ is defined by eqn: Bdef.

Theorems & Definitions (30)

  • Remark 2.1
  • Conjecture 2.2: CFKRS cfkrs
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Conjecture 2.6
  • Remark 2.7
  • Proposition 2.8
  • proof
  • ...and 20 more