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

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 $S{L}_2(\mathbb{Z})$, 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.

Corrects minor errors from published version

Cited by in corpus (1)

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