Zeta functions over zeros of Zeta functions and an exponential-asymptotic view of the Riemann HypothesisReport as inadecuate




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* Corresponding author 1 IPHT - Institut de Physique Théorique - UMR CNRS 3681

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.





Author: André Voros -

Source: https://hal.archives-ouvertes.fr/



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