Möbius orthogonality for the Zeckendorf sum-of-digits function

Abstract

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

15 pages