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Möbius orthogonality for q-semimultiplicative sequences

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Abstract

We show that all q-semimultiplicative sequences are asymptotically orthogonal to the Möbius function, thus proving the Sarnak conjecture for this class of sequences. This generalises analogous results for the sum-of-digits function and other digital sequences which follow from previous work of Mauduit and Rivat.

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Notes

  1. To justify the change of terminology, let us note that one would expect that a q-multiplicative sequence should be q-quasimultiplicative. This is true with the terminology introduced above, but not with the one in [34]. Similar remarks are made in [34, Section 1].

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Acknowledgements

The author is grateful to El Houcein El Abdalaoui, Tanja Eisner, Aihua Fan, Dominik Kwietniak, Imre Kátai, Mariusz Lemańczyk, Christian Mauduit, Stephan Wagner and Tamar Ziegler for helpful discussions and comments, and to the Referee for careful reading of the manuscript. The author is supported by ERC Grant ErgComNum 682150.

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  1. Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem Givat Ram, 9190401, Jerusalem, Israel

    Jakub Konieczny

  2. Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, Łojasiewicza 6, 30-348, Kraków, Poland

    Jakub Konieczny

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Konieczny, J. Möbius orthogonality for q-semimultiplicative sequences. Monatsh Math 192, 853–882 (2020). https://doi.org/10.1007/s00605-020-01435-2

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