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

作者：Hough Bob^*

来源：Forum Mathematicum, 2014, 26(2): 523-546.

DOI：10.1515/forum-2011-0105

• Zeta and L functions
• other Dirichlet series and zeta functions
• relations to random matrix theory

We consider the logarithm of the central value log L(1/2) in the orthogonal family {L(s, f)}(f is an element of Hk) where H-k is the set of weight k Hecke-eigen cusp forms for SL2(Z), and in the symplectic family {L(s, chi(8d))}(dxD) 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 -1/2 log log k and variance log log k, and in the second case of mean 1/2 log logD 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.

• 出版日期2014-3