Mobius Randomness and Dynamics

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The logarithmic Sarnak conjecture for ergodic weights (2017, background): “The class of “reasonable” sequences is expected to include all bounded “low complexity” sequences, and in this direction a precise conjecture that uses the language of dynamical systems was formulated by Sarnak in [62, 63]: Conjecture (Sarnak).”
Algebraic trace functions over the primes (2012, background): “338], and with Sarnak’s recent precise formulation in terms of orthogonality of the Möbius function against function with low complexity, in the sense of entropy (see [39]).”
Equivalence of the Logarithmically Averaged Chowla and Sarnak Conjectures (2016, background): “In [50], [51], Sarnak introduced the following related conjecture.”
The Möbius function and distal flows (2013, background): “The Möbius Disjointness Conjecture of the second author asserts that μ is linearly disjoint from every X whose entropy is 0 [18], [19].”
On invariant measures for B-free systems (2014, unknown): “Following [13], call a subset A ⊂ Z admissible (more precisely, B-admissible) if |A mod bi| < bi for each i ≥ 1.”