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Möbius orthogonality for the Zeckendorf sum-of-digits function


Authors: Michael Drmota, Clemens Müllner and Lukas Spiegelhofer
Journal: Proc. Amer. Math. Soc. 146 (2018), 3679-3691
MSC (2010): Primary 11A63, 11N37; Secondary 11B25, 11L03
DOI: https://doi.org/10.1090/proc/14015
Published electronically: May 24, 2018
MathSciNet review: 3825824
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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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Michael Drmota
Affiliation: Institut für Diskrete Mathematik und Geometrie TU Wien, Wiedner Hauptstr. 8–10, 1040 Wien, Austria
MR Author ID: 59890
Email: michael.drmota@tuwien.ac.at

Clemens Müllner
Affiliation: Institut für Diskrete Mathematik und Geometrie TU Wien, Wiedner Hauptstr. 8–10, 1040 Wien, Austria
Email: clemens.muellner@tuwien.ac.at

Lukas Spiegelhofer
Affiliation: Institut für Diskrete Mathematik und Geometrie TU Wien Wiedner Hauptstr. 8–10, 1040 Wien, Austria
MR Author ID: 1026565
Email: lukas.spiegelhofer@tuwien.ac.at

Keywords: Zeckendorf sum-of-digits function, Möbius randomness, morphic sequences
Received by editor(s): June 29, 2017
Published electronically: May 24, 2018
Additional Notes: All authors were supported by the Austrian Science Foundation FWF, project F5502-N26, which is a part of the Special Research Program “Quasi Monte Carlo Methods: Theory and Applications”. Moreover, the authors want to acknowledge support by the project MuDeRa (Multiplicativity, Determinism and Randomness), which is a joint project between the ANR (Agence Nationale de la Recherche) and the FWF (Austrian Science Fund). Furthermore, the authors want to thank Mariusz Lemańczyk for very helpful discussions.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2018 American Mathematical Society