Rishikesh | Searcharxiv

arXiv2012

A Recursion Formula for Moments of Derivatives of Random Matrix Polynomials

We give asymptotic formulae for random matrix averages of derivatives of characteristic polynomials over the groups USp(2N), SO(2N) and O^-(2N). These averages are used to predict the asymptotic formulae for moments of derivatives of L-functions which arise in number theory. Each formula gives the leading constant of the asymptotic in terms of determinants of hypergeometric functions. We find a differential recurrence relation between these determinants which allows the rapid computation of the (k+1)-st constant in terms of the k-th and (k-1)-st. This recurrence is reminiscent of a Toda lattice equation arising in the theory of \tau-functions associated with Painlev\'e differential equations.

math.NT↗

arXiv2012

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

We derive formulas for the terms in the conjectured asymptotic expansions of the moments, at the central point, of quadratic Dirichlet $L$-functions, $L(1/2,χ_d)$, 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 $\det (\binom{2k-i-λ_{k-i+1}}{2k-2j})$.

math.NT↗