Journal of the European Mathematical Society


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Volume 15, Issue 4, 2013, pp. 1343–1374
DOI: 10.4171/JEMS/394

Published online: 2013-05-09

Ergodic properties of square-free numbers

Francesco Cellarosi[1] and Yakov G. Sinai[2]

(1) University of Illinois at Urbana-Champaign, USA
(2) Princeton University, United States

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.

Keywords: Square-free numbers, correlation functions, dynamical systems with pure point spectrum, ergodicity

Cellarosi Francesco, Sinai Yakov: Ergodic properties of square-free numbers. J. Eur. Math. Soc. 15 (2013), 1343-1374. doi: 10.4171/JEMS/394