https://cea.hal.science/cea-01076225Voros, AndréAndréVorosIPHT - Institut de Physique Théorique - UMR CNRS 3681 - CEA - Commissariat à l'énergie atomique et aux énergies alternatives - Université Paris-Saclay - CNRS - Centre National de la Recherche ScientifiqueZeta functions over zeros of Zeta functions and an exponential-asymptotic view of the Riemann HypothesisHAL CCSD2014[MATH] Mathematics [math]De Laborderie, EmmanuelleY. Takei2014-10-21 15:10:562023-03-24 14:52:592014-10-21 15:13:42enConference papershttps://cea.hal.science/cea-01076225/documentapplication/pdf1We 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).