Introductory surveys

• Peter Sarnak. Three lectures on the Möbius function, randomness, and dynamics.
• Terry Tao. The Chowla conjecture and the Sarnak conjecture.

Surveys from a dynamical point of view

• Sébastien Ferenczi, Joanna Kułaga-Przymus, and Mariusz Lemańczyk. Sarnak’s conjecture: what’s new.
  • Handout summarizing important results about multiplicative functions appearing in Section 2 of the survey (compiled by Ethan Ackelsberg). [pdf]

Möbius disjointness for special classes of functions

• Harold Davenport. On some infinite series involving arithmetical functions (II). The Quarterly Journal of Mathematics, 8(1):313-320, 1937.
• Jean Bourgain, Peter Sarnak, and Tamar Ziegler. Disjointness of Moebius from horocycle flows. In From Fourier analysis and number theory to Radon transforms and geometry, Dev. Math. (Springer, New York) 67-83, 2013.
• Ben Green and Terence Tao. The Möbius function is strongly orthogonal to nilsequences. Annals of Mathematics, 175(2):541-566, 2012.
• Adam Kanigowski, Mariusz Lemańczyk, and Maksym Radziwiłł. Rigidity in dynamics and Möbius disjointness. Fundamenta Mathematicae, 255(3):309-336, 2021.
  

Entropy

• Michael Brin and Garrett Stuck. Introduction to Dynamical Systems.
• Peter Walters. An Introduction to Ergodic Theory.
• William Parry. Zero entropy of distal and related transformations. In Topological Dynamics (Symposium, Colorado State Univ., Ft. Collins, Colo., 1967) 383-389, 1968. [pdf]

Chowla conjecture

• Olivier Ramaré. Chowla’s conjecture: from the Liouville function to the Moebius function.

Kátai’s orthogonality criterion

• Imre Kátai. A remark on a theorem of H. Daboussi. Acta Mathematica Hungarica, 47:223-225, 1986.
• Nikos Frantzikinakis and Bernard Host. Higher order Fourier analysis of multiplicative functions and application. Journal of the American Mathematical Society, 30(1):67-157, 2017.
• Vitaly Bergelson, Joanna Kułaga-Przymus, Mariusz Lemańczyk, and Florian K. Richter. A generalization of Kátai’s orthogonality criterion with applications. Discrete and Continuous Dynamical Systems, 39(5):2581-2612, 2019.

Relationship between the Sarnak conjecture, the Chowla conjecture, and the Riemann hypothesis

• Davit Karagulyan. On certain aspects of the Möbius randomness principle. Colloquium Mathematicum, 157:231-250, 2019.
  • Slides by Sohail Farhangi. [pdf]