### 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 language | English (US) |
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Title of host publication | Chaos: Poincare Seminar 2010 |

Publisher | Birkhauser Boston |

Pages | 125-168 |

Number of pages | 44 |

Volume | 66 |

ISBN (Print) | 9783034806961 |

DOIs | |

State | Published - 2013 |

Event | 14th Poincare Seminar 2010: Chaos - Paris, France Duration: Jun 5 2010 → Jun 5 2010 |

### Other

Other | 14th Poincare Seminar 2010: Chaos |
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Country | France |

City | Paris |

Period | 6/5/10 → 6/5/10 |

### Fingerprint

### ASJC Scopus subject areas

- Astronomy and Astrophysics

### Cite this

*Chaos: Poincare Seminar 2010*(Vol. 66, pp. 125-168). Birkhauser Boston. https://doi.org/10.1007/978-3-0348-0697-8_4

**Quantum chaos, Random Matrix theory, and the Riemann ζ-function.** / Bourgade, Paul; Keating, Jonathan P.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Chaos: Poincare Seminar 2010.*vol. 66, Birkhauser Boston, pp. 125-168, 14th Poincare Seminar 2010: Chaos, Paris, France, 6/5/10. https://doi.org/10.1007/978-3-0348-0697-8_4

}

TY - GEN

T1 - Quantum chaos, Random Matrix theory, and the Riemann ζ-function

AU - Bourgade, Paul

AU - Keating, Jonathan P.

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84896787687&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84896787687&partnerID=8YFLogxK

U2 - 10.1007/978-3-0348-0697-8_4

DO - 10.1007/978-3-0348-0697-8_4

M3 - Conference contribution

SN - 9783034806961

VL - 66

SP - 125

EP - 168

BT - Chaos: Poincare Seminar 2010

PB - Birkhauser Boston

ER -