### 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 |

Subtitle of host publication | Poincare Seminar 2010 |

Publisher | Birkhauser Boston |

Pages | 125-168 |

Number of pages | 44 |

ISBN (Print) | 9783034806961 |

DOIs | |

State | Published - Jan 1 2013 |

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

### Publication series

Name | Progress in Mathematical Physics |
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Volume | 66 |

ISSN (Print) | 1544-9998 |

### Other

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

City | Paris |

Period | 6/5/10 → 6/5/10 |

### 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