Möbius orthogonality for the Zeckendorf sum-of-digits function

Drmota, Michael; Müllner, Clemens; Spiegelhofer, Lukas

doi:10.1090/proc/14015

Möbius orthogonality for the Zeckendorf sum-of-digits function
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by Michael Drmota, Clemens Müllner and Lukas Spiegelhofer PDF

Proc. Amer. Math. Soc. 146 (2018), 3679-3691 Request permission

Abstract:

We show that the (morphic) sequence $(-1)^{s_\varphi (n)}$ is asymptotically orthogonal to all bounded multiplicative functions, where $s_\varphi$ denotes the Zeckendorf sum-of-digits function. In particular we have $\sum _{n<N} (-1)^{s_\varphi (n)} \mu (n) = o(N)$, that is, this sequence satisfies the Sarnak conjecture.

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