### Abstract

For a fixed integer s≥.1, we estimate exponential sums with harmonic sums [InlineMediaObject not available: see fulltext.] individually and on average, where H _{s} (n) is computed modulo a prime p. These bounds are used to derive new results about various congruences modulo p involving H _{s} (n). For example, our estimates imply that for any >0, the set {H _{s} (n):n<p ^{1/2+ε}} is uniformly distributed modulo a sufficiently large p. We also show that every residue class λ can be represented as [InlineMediaObject not available: see fulltext.] with max{n _{ν} |ν=1,. . . , 7}≥p ^{1/2+ε} , and we obtain an asymptotic formula for the number of such representations. The same results hold also for the values B _{p} _{-} _{r} (n) of Bernoulli polynomials where r is fixed, complementing some results of W. L. Fouche.

Original language | English |
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Pages (from-to) | 855-865 |

Number of pages | 11 |

Journal | Mathematische Zeitschrift |

Volume | 253 |

Issue number | 4 |

DOIs | |

Publication status | Published - Aug 2006 |

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## Cite this

*Mathematische Zeitschrift*,

*253*(4), 855-865. https://doi.org/10.1007/s00209-006-0939-5