### Moebius orthogonality in dynamics

Ergodic Theory and Dynamical Systems Seminar

30th November 2017, 3:00 pm – 4:00 pm

Howard House, 4th Floor Seminar Room

In 2010 P. Sarnak formulated the following conjecture: All deterministic sequences are orthogonal to the classical Moebius function \mu. That is:

$$\lim_{N\to\infty}\frac1N\sum_{n\leq N}f(T^nx)\mu(n)=0$$

for each topological dynamical system $(X,T)$ of zero entropy and $f\in C(X)$, $x\in X$. The original (main) motivation for this conjecture was that the famous Chowla's conjecture on autocorrelations of $\mu$ implies Sarnak's conjecture. During my talk, I'd rather not be overviewing concrete instances of the validity of Sarnak's conjecture, but will concentrate on presenting it as one more evidence of deep connections between ergodic theory and number theory. In particular, I'll show to which extent, we know now about equivalence of Sarnak and Chowla's conjectures.

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