Characteristic polynomials, the Hybrid model, and the Ratios Conjecture

Andrew Pearce-crump

In the 1960s Shanks conjectured that the $\zeta\'(\rho)$, where $\rho$ is a non-trivial zero of zeta, is both real and positive in the mean. Conjecturing and proving this result has a rich history, but efforts to generalise it to higher moments have so far failed. Building on the work of Keating and Snaith using characteristic polynomials from Random Matrix Theory, the Hybrid model of Gonek, Hughes and Keating, and the Ratios Conjecture of Conrey, Farmer, and Zirnbauer, we have been able to produce new conjectures for the full asymptotics of higher moments of the derivatives of zeta. This is joint work with Chris Hughes.

https://mathtube.org/lecture/video/characteristic-polynomials-hybrid-model-and-ratios-conjecture

Mathtube- Analytic Aspects of L-functions and Applications to Number Theory