### Abstract

We examine the Linnik and Selberg Conjectures concerning sums of Kloosterman sums, in all its aspects (x, m and n). We correct the precise form of the Conjecture and establish an analogue of Kuznetzov’s 1/6 exponent in the mn aspect. This, perhaps somewhat surprisingly, is connected with the transional ranges associated with asymptotics of Bessel Functions of large order.

Original language | English (US) |
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Title of host publication | Progress in Mathematics |

Publisher | Springer Basel |

Pages | 619-635 |

Number of pages | 17 |

DOIs | |

State | Published - Jan 1 2009 |

### Publication series

Name | Progress in Mathematics |
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Volume | 270 |

ISSN (Print) | 0743-1643 |

ISSN (Electronic) | 2296-505X |

### All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory
- Geometry and Topology

### Keywords

- Kloosterman sums
- Kuznetzov formula
- Ramanujan-Selberg conjectures

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## Cite this

Sarnak, P., & Tsimerman, J. (2009). On Linnik and Selberg’s conjecture about sums of Kloosterman sums. In

*Progress in Mathematics*(pp. 619-635). (Progress in Mathematics; Vol. 270). Springer Basel. https://doi.org/10.1007/978-0-8176-4747-6_20