$6$-th Norm of a Steinhaus Chaos

Abstract

We prove that for the Steinhaus Random Variable $z(n)$ \[\mathbb{E}\left(\left|\sum_{n\in E_{N, m}}z(n)\right|^6\right)\asymp |E_{N, m}|^3 \text{ for } m\ll(\log\log N)^{\frac{1}{3}},\] where \[E_{N, m}:=\{1\leq n:\Omega(n)=m\}\] and $\Omega(n)$ denotes the number of prime factors of $N$.