Note on the Chowla Conjecture and the Discrete Fourier Transform

Let $x\ge 1$ be a large integer, and let $a_0<a_1<\cdots <a_{k-1}$ be a small fixed integer $k$-tuple, and let $\mu (n)\in \{-1,0,1\}$ be the periodic Mobius function. This note shows that discrete Fourier transform analysis produces a simple solution of the periodic Chowla conjecture. More precisely, it leads to an asymptotic formula of the form ${\sum }_{n\le x}\mu (n+a_0)\mu (n+a_1)\cdots \mu (n+a_{k-1})=O(x(\log x{)}^{-c})$, where $c>0$ is an arbitrary constant.