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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

Online
ISSN
1435-5337
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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

## Abstract

We consider the logarithm of the central value $logL\left(1/2\right)$ in the orthogonal family ${\left\{L\left(s,f\right)\right\}}_{f\in {H}_{k}}$ where Hk is the set of weight k Hecke-eigen cusp forms for ${\text{SL}}_{2}\left(ℤ\right)$, and in the symplectic family ${\left\{L\left(s,{\chi }_{8d}\right)\right\}}_{d\asymp D}$ 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 $-1/2loglogk$ 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.

MSC: 11M06; 11M41; 11M50

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,

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