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Integral moments of L-functions

Published online by Cambridge University Press:  22 June 2005

J. B. Conrey ,
D. W. Farmer ,
J. P. Keating ,
M. O. Rubinstein  and
N. C. Snaith
J. B. Conrey
Affiliation:
American Institute of Mathematics and Department of Mathematics, Oklahoma State University, Stillwater, OK 74078-0613, USA. E-mail: conrey@aimath.org
D. W. Farmer
Affiliation:
American Institute of Mathematics, 360 Portage Avenue, Palo Alto, CA 94306, USA. E-mail: farmer@aimath.org
J. P. Keating
Affiliation:
School of Mathematics, University of Bristol, Clifton, Bristol BS8 1TW, United Kingdom. E-mail: J.P.Keating@bristol.ac.uk, N.C.Snaith@bristol.ac.uk
M. O. Rubinstein
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada. E-mail: mrubinst@uwaterloo.ca
N. C. Snaith
Affiliation:
School of Mathematics, University of Bristol, Clifton, Bristol BS8 1TW, United Kingdom. E-mail: J.P.Keating@bristol.ac.uk, N.C.Snaith@bristol.ac.uk
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Abstract

We give a new heuristic for all of the main terms in the integral moments of various families of primitive $L$-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are defined by the appropriate group averages. This lends support to the idea that arithmetical $L$-functions have a spectral interpretation, and that their value distributions can be modelled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.

Keywords

11M2615A52
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 91 , Issue 1 , July 2005 , pp. 33 - 104
Copyright
2005 London Mathematical Society

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Footnotes

Research partially supported by the American Institute of Mathematics and a Focused Research Group grant from the National Science Foundation. The last author was also supported by a Royal Society Dorothy Hodgkin Fellowship.