On arithmetic functions orthogonal to deterministic sequences

Adam Kanigowski (1) , Joanna Kułaga-Przymus (2) , Mariusz Lemańczyk (2) , Thierry de La Rue (3)

1 Department of Mathematics [Maryland]
2 Faculty of Mathematics and Computer Science [Toruń]
3 LMRS - Laboratoire de Mathématiques Raphaël Salem

We prove Veech's conjecture on the equivalence of Sarnak's conjecture on Möbius orthogonality with a Kolmogorov type property of Furstenberg systems of the Möbius function. This yields a combinatorial condition on the Möbius function itself which is equivalent to Sarnak's conjecture. As a matter of fact, our arguments remain valid in a larger context: we characterize all bounded arithmetic functions orthogonal to all topological systems whose all ergodic measures yield systems from a fixed characteristic class (zero entropy class is an example of such a characteristic class). This allows us to show ergodically that bounded multiplicative functions with zero mean in arithmetic progressions on a typical short interval satisfy the averaged Chowla property established earlier by Matomäki, Radziwiłł and Tao.

Domaines: Systèmes dynamiques [math.DS]; Théorie des nombres [math.NT]

Adam Kanigowski, Joanna Kułaga-Przymus, Mariusz Lemańczyk, Thierry de La Rue. On arithmetic functions orthogonal to deterministic sequences. 2021. ⟨hal-03233409v1⟩