Helson's conjecture for smooth numbers

TL;DR

This paper proves that for Steinhaus random multiplicative functions, the partial sums over smooth numbers exhibit better than square-root cancellation uniformly across a wide range of parameters, indicating significant cancellation in these sums.

Contribution

It establishes that sums of Steinhaus random multiplicative functions over smooth numbers have stronger cancellation than previously known, uniformly for all y up to x.

Findings

Expected sum magnitude is o(Ψ(x,y)^{1/2})

Results hold uniformly for 2 ≤ y ≤ x

Provides quantitative bounds with large savings when y is not close to x

Abstract

Let $\Psi(x,y)$ 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 $$ \mathbb{E} \Big|\sum_{\substack{1\leq n \leq x\\ P(n) \leq y}} f(n) \Big| = o\left( \Psi(x,y)^{1/2} \right),$$ uniformly on the entire range $ 2 \leq y \leq x$. The bounds are quantitative and give a large saving when $y$ isn't too close to $x$.