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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

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Volume 26, Issue 2


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

Bob Hough
Published Online: 2012-01-13 | DOI: https://doi.org/10.1515/forum-2011-0105

An erratum for this article can be found here: https://doi.org/10.1515/forum-2014-0064


We consider the logarithm of the central value logL(1/2) in the orthogonal family {L(s,f)}fHk where Hk is the set of weight k Hecke-eigen cusp forms for SL2(), and in the symplectic family {L(s,χ8d)}dD where χ8d 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 -1/2loglogk and variance loglogk, and in the second case of mean 1/2loglogD and variance loglogD. 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.

Keywords: Zeta and L functions; other Dirichlet series and zeta functions; relations to random matrix theory

MSC: 11M06; 11M41; 11M50

About the article

Received: 2011-10-03

Revised: 2011-11-15

Published Online: 2012-01-13

Published in Print: 2014-03-01

Citation Information: Forum Mathematicum, Volume 26, Issue 2, Pages 523–546, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2011-0105.

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