Recent results of Katz and Sarnak [8, 9] suggest that the low-lying zeros of families of L-functions display the statistics of the eigenvalues of one of the compact groups of matrices U(N), O(N) or USp(2N). We here explore the link between the value distributions of the L-functions within these families at the central point s= 1/2 and those of the characteristic polynomials Z(U,θ) of matrices U with respect to averages over SO(2N) and USp(2N) at the corresponding point θ= 0, using techniques previously developed for U(N) in [10]. For any matrix size N we find exact expressions for the moments of Z(U,0) for each ensemble, and hence calculate the asymptotic (large N) value distributions for Z(U,0) and log Z(U,0). The asymptotic results for the integer moments agree precisely with the few corresponding values known for L-functions. The value distributions suggest consequences for the non-vanishing of L-functions at the central point.

http://www.hpl.hp.com/techreports/2000/HPL-BRIMS-2000-05.pdf

Random matrix theory and L-functions at s=1/2

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.

https://web.williams.edu/Mathematics/sjmiller/public_html/ntandrmt/handouts/conrey/ConreyFarmerKeatingRubinsteinSnaith_IntegralMomentsLFns.pdf

Integral moments of L-functions

Modular forms Modular forms of level $1$ Modular forms of weight $2$ Dirichlet characters Eisenstein series and Bernoulli numbers Dimension formulas Linear algebra General modular symbols Computing with newforms Computing periods Solutions to selected exercises Appendix A: Computing in higher rank Bibliography Index.

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Modular forms, a computational approach

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 modeled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.

To the general mathematician L-functions might appear to be an esoteric and special topic in number theory. We hope that the discussion below will convince the reader otherwise. Time and again it has turned out that the crux of a problem lies in the theory of these functions. At some level it is not entirely clear to us why L-functions should enter decisively, though in hindsight one can give reasons. Our plan is to introduce L-functions and describe the central problems connected with them. We give a sample (this is certainly not meant to be a survey) of results towards these conjectures as well as some problems that can be resolved by finessing these conjectures. We also mention briefly some of the successful present-day tools and the role they might play in the big picture.

Perspectives on the Analytic Theory of L-Functions

We give a new heuristic for all of the main terms in the quotient of products of L-functions averaged over a family. These conjectures generalize the recent conjectures for mean values of L-functions. Comparison is made to the analogous quantities for the characteristic polynomials of matrices averaged over a classical compact group.

http://www.intlpress.com/site/pub/files/_fulltext/journals/cntp/2008/0002/0003/CNTP-2008-0002-0003-a004.pdf

Autocorrelation of ratios of $L$-functions

We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than large-matrix asymptotic approximations. They also mirror exactly the autocorrelation formulae conjectured to hold for L-functions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of L-functions.

Autocorrelation of Random Matrix Polynomials

We conjecture the true rate of growth of the maximum size of the Riemann zeta-function and other L-functions. We support our conjecture using arguments from random matrix theory, conjectures for moments of L-functions, and also by assuming a random model for the primes.

David W. Farmer

Papers

Integral moments of L-functions

Integral moments of L-functions

Autocorrelation of ratios of $L$-functions

Autocorrelation of Random Matrix Polynomials

The maximum size of L-functions