Quantum chaos, Random Matrix theory, and the Riemann ζ-function

Paul Bourgade, Jonathan P. Keating

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We review some connections between quantum chaos and the theory of the Riemann zeta function and the primes. Specifically, we give an overview of the similarities between the semiclassical trace formula that connects quantum energy levels and classical periodic orbits in chaotic systems and an analogous formula that connects the Riemann zeros and the primes. We also review the role played by Random Matrix Theory in both quantum chaos and the theory of the zeta function. The parallels we review are conjectural and still far from being understood, but the ideas have led to substantial progress in both areas.

Original languageEnglish (US)
Title of host publicationChaos
Subtitle of host publicationPoincare Seminar 2010
PublisherBirkhauser Boston
Pages125-168
Number of pages44
ISBN (Print)9783034806961
DOIs
StatePublished - Jan 1 2013
Event14th Poincare Seminar 2010: Chaos - Paris, France
Duration: Jun 5 2010Jun 5 2010

Publication series

NameProgress in Mathematical Physics
Volume66
ISSN (Print)1544-9998

Other

Other14th Poincare Seminar 2010: Chaos
CountryFrance
CityParis
Period6/5/106/5/10

ASJC Scopus subject areas

  • Astronomy and Astrophysics

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  • Cite this

    Bourgade, P., & Keating, J. P. (2013). Quantum chaos, Random Matrix theory, and the Riemann ζ-function. In Chaos: Poincare Seminar 2010 (pp. 125-168). (Progress in Mathematical Physics; Vol. 66). Birkhauser Boston. https://doi.org/10.1007/978-3-0348-0697-8_4