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**Some applications of the large sieve in Riemann surfaces.**
*(English)*
Zbl 0863.11062

We present some consequences of large sieve type inequalities related to the spectral analysis in a Riemann surface. We first prove some average results for the hyperbolic circle problem. For some special Fuchsian groups they imply average results for some arithmetical functions, for instance, for \(\sum_{k\leq n} r(k) r(k+ l)\) where \(r(k)\) is the number of representations as a sum of two squares or for the number of integral matrices with bounded entries leaving \(x^2- y^2- z^2\) invariant. We also show results relating the arithmetical character of the involved groups to spectral theory. Finally, a general large sieve inequality in compact manifolds is applied to get an average result for the off-centered (classical) circle problem and it is also applied in a physical context. [See also the author’s paper in Acta Arith. 77, 303–313 (1996; Zbl 0863.11061) reviewed above].

Reviewer: Fernando Chamizo (Madrid)

### MSC:

11N35 | Sieves |

11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |

11F25 | Hecke-Petersson operators, differential operators (one variable) |

30F99 | Riemann surfaces |

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |