cea-01076225
https://cea.hal.science/cea-01076225
https://cea.hal.science/cea-01076225/document
https://cea.hal.science/cea-01076225/file/1403.4558v2.pdf
arxiv:1403.4558
[CEA] CEA - Commissariat à l'énergie atomique
[CNRS] CNRS - Centre national de la recherche scientifique
[INSMI] CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions
[DSM-IPHT] IPHT
[CEA-DRF] Direction de Recherche Fondamentale
[GS-MATHEMATIQUES] Graduate School Mathématiques
[GS-PHYSIQUE] Graduate School Physique
Zeta functions over zeros of Zeta functions and an exponential-asymptotic view of the Riemann Hypothesis
Voros, André
[MATH] Mathematics [math]
COMM
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).
2013-10-15
2014-10-21
en
RIMS Kôkyûroku Bessatsu
RIMS, Kyoto University
Exponential analysis of differential equations and related topics
Kyoto, Japan