Note on the Chowla Conjecture and the Discrete Fourier Transform

Let $x\geq 1$ 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 $\sum_{n \leq x} \mu(n+a_0) \mu(n+a_1)\cdots\mu(n+a_{k-1}) =O\left( x(\log x)^{-c}\right)$, where $c>0$ is an arbitrary constant.
Comment: Thirty Four Pages. Keywords: Arithmetic function; Mobius function; Liouville function; Autocorrelation function; Correlation function; Chowla conjecture; Discrete Fourier transform