Zeta functions over zeros of Zeta functions and an exponential-asymptotic view of the Riemann Hypothesis : dedicated to Professor Takashi AOKI for his 60th birthday (Exponential Analysis of Differential Equations and Related Topics)

Abstract

We review generalized zeta functions built over the Riemann zeros (in short: "superzeta" functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz zeta function. As a concrete application, a superzeta function enters an integral representation for the KeiperLi coefficients, whose large-order behavior thereby becomes computable by the method of steepest descents; then the dominant saddle-point entirely depends on the Riemann Hypothesis being true or not, and the outcome is a sharp exponential-asymptotic criterion for the Riemann Hypothesis that only refers to the large-order KeiperLi coefficients. As a new result, that criterion, then Li' s criterion, are transposed to a novel sequence of Riemann-zeta expansion coefficients based at the point 1/2 (vs 1 for KeiperLi).