HAL CCSD
Zeta functions over zeros of Zeta functions and an exponential-asymptotic view of the Riemann Hypothesis
Voros, André
Institut de Physique Théorique - UMR CNRS 3681 (IPHT) ; Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)
International audience
RIMS Kôkyûroku Bessatsu
Exponential analysis of differential equations and related topics
Kyoto, Japan
Y. Takei
Y. Takei
RIMS, Kyoto University
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
https://cea.hal.science/cea-01076225
Exponential analysis of differential equations and related topics, Y. Takei, Oct 2013, Kyoto, Japan. pp.147-164
ARXIV: 1403.4558
info:eu-repo/semantics/altIdentifier/arxiv/1403.4558
en
MSC 11-02, 11-06, 11M26, 11M41, 11M35, 30B40, 30E15, 41A60
[MATH]Mathematics [math]
info:eu-repo/semantics/conferenceObject
Conference papers
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
info:eu-repo/semantics/OpenAccess