Note on the Chowla Conjecture and the Discrete Fourier Transform

TL;DR

This paper demonstrates that discrete Fourier transform techniques can be used to prove a form of the periodic Chowla conjecture, providing an asymptotic estimate for sums involving the Möbius function.

Contribution

It introduces a simple Fourier analysis approach to establish an asymptotic formula related to the periodic Chowla conjecture, advancing understanding of Möbius function correlations.

Findings

Derived an asymptotic formula for Möbius function sums

Showed Fourier analysis as a tool for Chowla conjecture

Established bounds with arbitrary decay rate

Abstract

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.