Speaker：Michael Drmota（TU Wien）

Time：2022-10-27, 15:00

Location：Tencent Meeting ID：947-841-4036（Password：260172）

Abstract:

A sequence a((n)) of complex numbers is said to be Moebius orthogonal if \sum_{n\le N} a(n)\mu(n)=o(N) as N\to \infty, where \mu(n) denotes the Moebius function (defined by \mu(p_1,p_2,\ldots,p_k))=(-1)^k for different prime numbers p_1,\ldots,p_k and \mu(n)=0 else). For example, the prime number theorem is equivalent to the fact that a(n)=1 is Moebius orthogonal, and the Dirichlet prime number theorem is equivalent to the fact that periodic sequences a(n) are Moebius orthogonal. It is therefore a natural question to ask which sequences a(n) are Moebius orthogonal. Recently Peter Sarnak formulated a concrete conjecture on this question, namely that sequences of the form a(n)=f(T^n, x_0) that are generated by a discrete dynamical system (X,T) of zero entropy and a continuous function f: X \to C are Moebius orthogonal. He also showed that this property is implied by the famous Chowla conjecture. In the mean time the Sarnak conjecture has been verified for several classes of sequences, for example for nilsequences or for authomatic sequences.

The purpose of this talk is to present some recent developments on the Sarnak conjecture and related questions, where the sequence a(n) is related to various enumeration systems or substitution systems. Since these kind of sequences rely -- more or less—on the additive structure of the integers, the verification of the Sarnak conjecture in this context can be also seen as a result on the independence of addition and multiplication of integers.