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

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 ) $\log L(1/2)$ in the orthogonal family { L ( s , f ) } f ∈ H k $\lbrace L(s,f)\rbrace _{f \in H_k}$ where H k is the set of weight k Hecke-eigen cusp forms for SL 2 ( ℤ ) $\text{SL}_2(\mathbb {Z})$ , and in the symplectic family { L ( s , χ 8 d ) } d ≍ D $\lbrace L(s,\chi _{8d})\rbrace _{d \asymp D}$ where χ 8 d $\chi _{8d}$ is the real character associated to fundamental discriminant 8 d $8d$ . 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 $-1/2 \log \log k$ and variance log log k $\log \log k$ , and in the second case of mean 1 / 2 log log D $1/2\log \log D$ and variance log log D $\log \log D$ . 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

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

Hough, Bob
Forum Mathematicum , Volume 26 (2) – Mar 1, 2014
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Publisher
de Gruyter
Copyright
Copyright © 2014 by the
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2011-0105
Publisher site
See Article on Publisher Site

Abstract

Abstract We consider the logarithm of the central value log L ( 1 / 2 ) $\log L(1/2)$ in the orthogonal family { L ( s , f ) } f ∈ H k $\lbrace L(s,f)\rbrace _{f \in H_k}$ where H k is the set of weight k Hecke-eigen cusp forms for SL 2 ( ℤ ) $\text{SL}_2(\mathbb {Z})$ , and in the symplectic family { L ( s , χ 8 d ) } d ≍ D $\lbrace L(s,\chi _{8d})\rbrace _{d \asymp D}$ where χ 8 d $\chi _{8d}$ is the real character associated to fundamental discriminant 8 d $8d$ . 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 $-1/2 \log \log k$ and variance log log k $\log \log k$ , and in the second case of mean 1 / 2 log log D $1/2\log \log D$ and variance log log D $\log \log D$ . 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.

Journal

Forum Mathematicum – de Gruyter

Published: Mar 1, 2014

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