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On Arithmetic Properties of Products and Shifted Products

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Analytic Number Theory

Abstract

Let \(\mathcal{A}\) and \(\mathcal{B}\) be large subsets of \(\{1\ldots,N\}\). Arithmetic properties of the products ab, resp. of the shifted products ab + 1 with \(a \in \mathcal{A}\), \(b \in \mathcal{B}\) are studied. In particular, it is shown that the sum of digits of the products ab is well distributed, and the size of the subsets \(\mathcal{A},\mathcal{B}\in \{ 1\ldots,N\}\) with the property that ab + 1 is always squarefree is estimated.

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Acknowledgements

Research partially supported by the Hungarian National Foundation for Scientific Research, Grants No K100291 and NK104183, the French-Hungarian Balaton exchange program FR-33/2009 and the Agence Nationale de la Recherche grant ANR-10-BLAN 0103 MUNUM.

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Authors and Affiliations

  1. Université d’Aix-Marseille, Institut de Mathématiques de Marseille, CNRS UMR 7373, 163 avenue de Luminy, Case 907, 13288, Marseille Cedex 9, France

    Joël Rivat

  2. Department of Algebra and Number Theory, Eötvös Loránd University, H-1117, Budapest, Pázmány Péter sétány 1/c, Hungary

    András Sárközy

Authors
  1. Joël Rivat
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  2. András Sárközy
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Correspondence to Joël Rivat .

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Editors and Affiliations

  1. Department of Mathematics, Dartmouth College, Hanover, New Hampshire, USA

    Carl Pomerance

  2. Department of Mathematics, ETH-Zürich, Zürich, Switzerland

    Michael Th. Rassias

Additional information

Dedicated to Helmut Maier on the occasion of his 60th birthday

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Rivat, J., Sárközy, A. (2015). On Arithmetic Properties of Products and Shifted Products. In: Pomerance, C., Rassias, M. (eds) Analytic Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-22240-0_21

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