# Workshop on Dynamics, Discrete Analysis and Multiplicative Number Theory

## On a Conjecture of Veech About Möbius Orthogonality

Abstract: In unpublished lecture notes, William A. Veech considered the following potential property of the Möbius function:

"In any Furstenberg system of the Möbius function, the zero-coordinate is orthogonal to any function measurable with respect to the Pinsker sigma-algebra of the system."

Veech proved that this property would imply the validity of Sarnak's conjecture, that is orthogonality of the Möbius function to any topological dynamical system of zero entropy. He also conjectured that both properties are equivalent.

In this talk based on a joint work Adam Kanigowski, Joanna Kułaga-Przymus and Mariusz Lemańczyk, I will sketch the proof of this equivalence. Our proof is in fact valid in a more general setting.

We introduce the notion of "characteristic classes" of measure-theoretic dynamical systems, which are classes stable by taking factors and joinings. (The class of systems whose Kolmogorov-Sinai entropy vanishes is an example of such a characteristic class.) The equivalence conjectured by Veech remains true if we replace the zero-entropy property by any class of topological dynamical systems whose visible invariant measures remain in a given characteristic class, provided the Pinsker sigma-algebra is also replaced by the maximum factor sigma-algebra in this characteristic class. The Möbius function can also be replaced by any bounded arithmetic function.

I will also discuss the combinatorial property of the arithmetic function that follows from the Veech condition.