Helson's conjecture for smooth numbers

Abstract

Let $\hat{I}¨(x,y)$ denote the count of $y$-smooth numbers below $x$ and $P(n)$ denote the largest prime factor of $n$. We prove that for $f$ a Steinhaus random multiplicative function, the partial sums over $y$-smooth numbers always enjoy better than squareroot cancellation, in the sense that $$\mathbb{E}|\sum _{\begin{array}{c}1\le n\le x\\ P(n)\le y\end{array}}f(n)|=o\left(\hat{I}¨(x,y{)}^{1/2}\right),$$ uniformly on the entire range $2\le y\le x$. The bounds are quantitative and give a large saving when $y$ isn't too close to $x$.

35 pages, including an 11-page introduction. This is a significant update which simplifies the proof of Theorem 1.2 (previously Theorem 1.3) and covers the whole range of smoothness parameter