The distribution of the logarithm in an orthogonal and a symplectic family of $L$-functions

TL;DR

This paper studies the distribution of the logarithm of central values of $L$-functions in orthogonal and symplectic families, showing they are asymptotically Gaussian and confirming a conjecture under certain hypotheses.

Contribution

It proves the asymptotic Gaussian distribution of $\log L(1/2)$ in two families of $L$-functions, unconditionally bounded above and conditionally following a normal law.

Findings

Distributions are asymptotically bounded above by Gaussian distributions.

Conditional on hypotheses, the distributions follow a full normal law.

Results confirm a conjecture of Keating and Snaith.

Abstract

We consider the logarithm of the central value $\log L(1/2)$ in the orthogonal family ${L(s,f)}_{f \in H_k}$ where $H_k$ is the set of weight $k$ Hecke-eigen cusp form for $SL_2(\mathbb{Z})$, and in the symplectic family ${L(s,\chi_{8d})}_{d \asymp D}$ where $\chi_{8d}$ is the real character associated to fundamental discriminant $8d$. Unconditionally, we prove that the two distributions are asymptotically bounded above by Gaussian distributions, in the first case of mean $-\frac{1}{2} \log \log k$ and variance $\log \log k$, and in the second case of mean $\frac{1}{2}\log \log D$ and variance $\log \log D$. Assuming both the Riemann and Zero Density Hypotheses in these families we obtain the full normal law in both families, confirming a conjecture of Keating and Snaith.