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Distribution modulo 1 of a linear sequence associated to a multiplicative function evaluated at polynomial arguments

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Abstract

In two previous papers, the first named author jointly with Florian Luca and Henryk Iwaniec, have studied the distribution modulo 1 of sequences which have linear growth and are mean values of multiplicative functions on the set of all the integers. In this note, we give a first result concerning sequences with linear growth associated to the mean values of multiplicative functions on a set of polynomial values, proving the density modulo 1 of the sequence

$$ \left( {\sum\limits_{m \leqslant n} {\frac{{\phi (m^2 + 1)}} {{m^2 + 1}}} } \right)_n . $$

This result is but an illustration of the theme which is currently being developed in the PhD thesis of the second named author.

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References

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Authors and Affiliations

  1. Institut Mathématique de Bordeaux, UMR 5251, Université de Bordeaux et CNRS, 33405, TALENCE Cedex, France

    Jean-Marc Deshouillers & Mehdi Hassani

  2. Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, 45137-66731, P.O. Box 45195-1159, Iran

    Mehdi Hassani

Authors
  1. Jean-Marc Deshouillers
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  2. Mehdi Hassani
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Corresponding author

Correspondence to Jean-Marc Deshouillers.

Additional information

Dedicated to Professor Wang Yuan on the Occasion of his 80th Birthday

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Deshouillers, JM., Hassani, M. Distribution modulo 1 of a linear sequence associated to a multiplicative function evaluated at polynomial arguments. Sci. China Math. 53, 2203–2206 (2010). https://doi.org/10.1007/s11425-010-4036-1

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  • DOI: https://doi.org/10.1007/s11425-010-4036-1

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