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Proceedings of the London Mathematical Society 2005 91(1):33-104; doi:10.1112/S0024611504015175
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© London Mathematical Society

Integral Moments of L-Functions

J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein and N. C. Snaith

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

Received 11 November 2002. Revision received 9 June 2004.

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. 2000 Mathematics Subject Classification 11M26, 15A52.

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.


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