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On the mean values of L ‐functions in orthogonal and symplectic families
Author(s) -
Bui H. M.,
Keating J. P.
Publication year - 2008
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdm046
Subject(s) - mathematics , symplectic geometry , pure mathematics , dirichlet distribution , euler's formula , analytic number theory , class number formula , random matrix , riemann zeta function , arithmetic zeta function , eigenvalues and eigenvectors , mathematical analysis , dirichlet series , physics , quantum mechanics , boundary value problem
Abstract Hybrid Euler–Hadamard products have previously been studied for the Riemann zeta function on its critical line and for Dirichlet L ‐functions, in the context of the calculation of moments and connections with Random Matrix Theory. According to the Katz–Sarnak classification, these are believed to represent families of L ‐function with unitary symmetry. We here extend the formalism to families with orthogonal and symplectic symmetry. Specifically, we establish formulae for real quadratic Dirichlet L ‐functions and for the L ‐functions associated with primitive Hecke eigenforms of weight 2 in terms of partial Euler and Hadamard products. We then prove asymptotic formulae for some moments of these partial products and make general conjectures based on results for the moments of characteristic polynomials of random matrices.