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
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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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Additional Information
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