JournalsjemsVol. 15, No. 4pp. 1343–1374

Ergodic properties of square-free numbers

  • Francesco Cellarosi

    University of Illinois at Urbana-Champaign, USA

  • Yakov G. Sinai

    Princeton University, United States
Ergodic properties of square-free numbers cover
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Abstract

We construct a natural invariant measure concentrated on the set of square-free numbers, and invariant under the shift. We prove that the corresponding dynamical system is isomorphic to a translation on a compact, Abelian group. This implies that this system is not weakly mixing and has zero measure-theoretical entropy.

Cite this article

Francesco Cellarosi, Yakov G. Sinai, Ergodic properties of square-free numbers. J. Eur. Math. Soc. 15 (2013), no. 4, pp. 1343–1374

DOI 10.4171/JEMS/394