The Möbius function and distal flows. (English) Zbl 1383.11094

Liu, Jianya; Sarnak, Peter

Duke Math. J. 164, No. 7, 1353-1399 (2015).

Summary: We prove that the Möbius function is linearly disjoint from an analytic skew product on the 2-torus. These flows are distal and can be irregular in the sense that their ergodic averages need not exist for all points. The previous cases for which such disjointness has been proved are all regular. We also establish the linear disjointness of the Möbius function from various distal homogeneous flows.

MSC:

11L03	Trigonometric and exponential sums (general theory)
37A45	Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
11N37	Asymptotic results on arithmetic functions

Keywords:

Möbius function; distal flow; affine linear map; skew product; nilmanifold

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