J B Conrey

Integral moments of L -functions

We give a new heuristic for all of the main terms in the integral moments of various families of primitive

Averages of ratios of characteristic polynomials for the compact classical groups

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Discretisation for Odd Quadratic Twists

-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

Applications of the L-functions ratios conjectures

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Mean values of L-functions and symmetry

-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.

On the frequency of vanishing of quadratic twists of modular L-functions

Averages of ratios of characteristic polynomials for the compact classical groups are evaluated in terms of determinants whose dimensions are independent of the matrix rank. These formulas are shown to be equivalent to expressions for the same averages obtained in a previous study, which was motivated by applications to analytic number theory. Our approach uses classical methods of random matrix theory, in particular determinants and orthogonal polynomials, and can be considered more elementary than the method of Howe pairs used in the previous study.

Lower order terms in the full moment conjecture for the Riemann zeta function

The discretisation problem for even quadratic twists is almost understood, with the main question now being how the arithmetic Delaunay heuristic interacts with the analytic random matrix theory prediction. The situation for odd quadratic twists is much more mysterious, as the height of a point enters the picture, which does not necessarily take integral values (as does the order of the Shafarevich-Tate group). We discuss a couple of models and present data on this question.