Helson's conjecture for smooth numbers
Seth Hardy, Max Wenqiang Xu
TL;DR
This work studies Helson-type cancellation for partial sums of a Steinhaus random multiplicative function over $y$-smooth numbers, establishing that $\mathbb E\bigl|\sum_{n\le x, P(n)\le y} f(n)\bigr| = o\bigl(\Psi(x,y)^{1/2}\bigr)$ uniformly for $(\log x)^{30} \le y \le x$. The authors develop a two-regime framework: near $y\approx x$ the cancellation arises from Gaussian multiplicative chaos features of the Euler product, while for moderately smooth $y$ they exploit conditioning on primes and refined Euler-product moment estimates to achieve substantial savings beyond squareroot cancellation. Central to the approach are saddle-point analyses of $y$-smooth counts, random Euler products evaluated on shifted lines $\Re s = \alpha/2$, and a Plancherel-based identity linking first moments to averages of Euler products. The results yield explicit, uniform bounds across two regimes: (i) large $y$ with GMC-driven savings and (ii) moderately smooth $y$ with bounds in terms of $\Psi(x^2,y)$ and $u=\frac{\log x}{\log y}$, including quantified reductions by factors like $\exp(O(u^{8/11}))$. These findings contribute to understanding smoother-than-squareroot cancellation and have potential applications to counting $y$-smooth numbers in short intervals and related questions in multiplicative number theory.
Abstract
Let $Ψ(x,y)$ 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 enjoy better than squareroot cancellation, in the sense that $$ \mathbb E \Big|\sum_{\substack{1\leq n \leq x\\ P(n) \leq y}} f(n) \Big| = o\left( Ψ(x,y)^{1/2} \right),$$ uniformly for $(\log x)^{30} \leq y \leq x$. Our bounds are quantitative and give a large saving when $y$ isn't too close to $x$.