Ergodic properties of square-free numbers

  • Yakov G. Sinai

    Princeton University, United States
  • Francesco Cellarosi

    University of Illinois at Urbana-Champaign, USA


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

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

DOI 10.4171/JEMS/394