Zeta functions over zeros of Zeta functions and an exponential-asymptotic view of the Riemann Hypothesis

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 repre-sentation for the Keiper–Li coefficients, whose large-order behavior thereby becomes computable by the method of steepest descents; then the dominant saddle-point en-tirely 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 Keiper–Li 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 Keiper–Li).
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  • ARXIV : 1403.4558

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André Voros. Zeta functions over zeros of Zeta functions and an exponential-asymptotic view of the Riemann Hypothesis. Exponential analysis of differential equations and related topics, Y. Takei, Oct 2013, Kyoto, Japan. pp.147-164. ⟨cea-01076225⟩

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