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We show that all $q$-semimultiplicative sequences are asymptoticallyorthogonal to the M\"obius function, thus proving the Sarnak conjecture forthis class of sequences. This generalises analogous results for thesum-of-digits function and other digital sequences which follow from previouswork of Mauduit and Rivat.

M\"obius orthogonality for $q$-semimultiplicative sequences