Conjectures and experiments concerning the moments of $L(1/2,\chi_d)$

TL;DR

This paper presents extensive numerical experiments on the moments of quadratic Dirichlet L-functions at the critical point, testing conjectured asymptotics and lower order terms through large-scale computations.

Contribution

It provides the first large-scale numerical verification of conjectured moments and lower order terms for quadratic Dirichlet L-functions at the critical point.

Findings

Numerical data supports the conjectured asymptotics for moments.

Evidence for lower order terms in the moments matches theoretical predictions.

Algorithms developed enable large-scale computations of L-values.

Abstract

We report on some extensive computations and experiments concerning the moments of quadratic Dirichlet $L$-functions at the critical point. We computed the values of $L(1/2,\chi_d)$ for $- 5\times 10^{10} < d < 1.3 \times 10^{10}$ in order to numerically test conjectures concerning the moments $\sum_{|d|<X} L(1/2,\chi_d)^k$. Specifically, we tested the full asymptotics for the moments conjectured by Conrey, Farmer, Keating, Rubinstein, and Snaith, as well as the conjectures of Diaconu, Goldfeld, Hoffstein, and Zhang concerning additional lower terms in the moments. We also describe the algorithms used for this large scale computation.