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

An erratum for this article can be found here: https://doi.org/10.1515/forum-2014-0064
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Published/Copyright: January 13, 2012
Forum Mathematicum
From the journal Forum Mathematicum Volume 26 Issue 2

Abstract

We consider the logarithm of the central value logL(1/2) in the orthogonal family {L(s,f)}f∈Hk where Hk is the set of weight k Hecke-eigen cusp forms for SL2(ℤ), and in the symplectic family {L(s,χ8d)}d≍D 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
Received: 2011-10-3
Revised: 2011-11-15
Published Online: 2012-1-13
Published in Print: 2014-3-1

© 2015 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Centralizer of an idempotent in a reductive monoid
  3. Free subgroups within the images of quantum representations
  4. The Bergman property for endomorphism monoids of some Fraïssé limits
  5. Boundary value problems for heat equation with certain potentials in Lipschitz cylinders
  6. On some multiplicity and mixed multiplicity formulas
  7. Free group algebras in Malcev–Neumann skew fields of fractions
  8. Decomposition theorems for QK spaces and applications
  9. Ideals of generalized invariants, local cohomology and Macaulay's double duality theorem
  10. The distribution of the logarithm in an orthogonal and a symplectic family of L-functions
  11. On seven-dimensional quaternionic contact solvable Lie groups
  12. Weitzenböck derivations of nilpotency 3
  13. Unital associahedra
  14. On the norm of the Lp-Fourier transform on compact extensions of ℝn
  15. On the existence of semifields of order q8 admitting elementary abelian groups of order 8
https://doi.org/10.1515/forum-2011-0105
Keywords for this article
Zeta and L functions; other Dirichlet series and zeta functions; relations to random matrix theory

Articles in the same Issue

  1. Frontmatter
  2. Centralizer of an idempotent in a reductive monoid
  3. Free subgroups within the images of quantum representations
  4. The Bergman property for endomorphism monoids of some Fraïssé limits
  5. Boundary value problems for heat equation with certain potentials in Lipschitz cylinders
  6. On some multiplicity and mixed multiplicity formulas
  7. Free group algebras in Malcev–Neumann skew fields of fractions
  8. Decomposition theorems for QK spaces and applications
  9. Ideals of generalized invariants, local cohomology and Macaulay's double duality theorem
  10. The distribution of the logarithm in an orthogonal and a symplectic family of L-functions
  11. On seven-dimensional quaternionic contact solvable Lie groups
  12. Weitzenböck derivations of nilpotency 3
  13. Unital associahedra
  14. On the norm of the Lp-Fourier transform on compact extensions of ℝn
  15. On the existence of semifields of order q8 admitting elementary abelian groups of order 8
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