### Abstract

A theory of Poincaré series is developed for Lobachevsky space of arbitrary dimension. For a general non-uniform lattice a Selberg-Kloosterman zeta function is introduced. It has meromorphic continuation to the plane with poles at the corresponding automorphic spectrum. When the lattice is a unit group of a rational quadratic form, the Selberg-Kloosterman zeta function is computed explicitly in terms of exponential sums. In this way a non-trivial Ramanujan-like bound analogous to "Selberg's 3/16 bound" is proved in general.

Original language | English (US) |
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Pages (from-to) | 231-237 |

Number of pages | 7 |

Journal | Proceedings of the Indian Academy of Sciences - Mathematical Sciences |

Volume | 97 |

Issue number | 1-3 |

DOIs | |

State | Published - Dec 1 1987 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- Lobachevsky space
- Poincaré series
- Selberg-Kloosterman zeta function
- non-uniform lattices

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## Cite this

Li, J. S., Piatetski-Shapiro, I., & Sarnak, P. (1987). Poincaré series for SO(n, 1).

*Proceedings of the Indian Academy of Sciences - Mathematical Sciences*,*97*(1-3), 231-237. https://doi.org/10.1007/BF02837825