Uniform asymptotics for the full moment conjecture of the Riemann zeta function

Conrey, Farmer, Keating, Rubinstein, and Snaith, recently conjectured formulas for the full asymptotics of the moments of $L$-functions. In the case of the Riemann zeta function, their conjecture states that the $2k$-th absolute moment of zeta on the critical line is asymptotically given by a certain $2k$-fold residue integral. This residue integral can be expressed as a polynomial of degree $k^2$, whose coefficients are given in exact form by elaborate and complicated formulas. In this article, uniform asymptotics for roughly the first $k$ coefficients of the moment polynomial are derived. Numerical data to support our asymptotic formula are presented. An application to bounding the maximal size of the zeta function is considered.Comment: 53 pages, 1 figure, 2 table

Secondary terms in the number of vanishings of quadratic twists of elliptic curve L-functions

We examine the number of vanishings of quadratic twists of the L-function associated to an elliptic curve. Applying a conjecture for the full asymptotics of the moments of critical L-values we obtain a conjecture for the first two terms in the ratio of the number of vanishings of twists sorted according to arithmetic progressions.Comment: 16 pages, many figure

Lower order terms for the moments of symplectic and orthogonal families of LL-functions

We derive formulas for the terms in the conjectured asymptotic expansions of the moments, at the central point, of quadratic Dirichlet $L$-functions, $L(1/2,\chi_d)$, and also of the $L$-functions associated to quadratic twists of an elliptic curve over \Q. In so doing, we are led to study determinants of binomial coefficients of the form $\det (\binom{2k-i-\lambda_{k-i+1}}{2k-2j})$.Comment: 34 pages, 4 table

Discretisation for odd quadratic twists

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.Comment: To appear in the Proceedings of the INI Workshop on Random Matrix Theory and Elliptic Curve