Helson's conjecture for smooth numbers
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912883700400128 |
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| author | Hardy, Seth Xu, Max Wenqiang |
| author_facet | Hardy, Seth Xu, Max Wenqiang |
| contents | 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 always enjoy better than squareroot cancellation, in the sense that $$ \mathbb{E} \Big|\sum_{\substack{1\leq n \leq x\\ P(n) \leq y}} f(n) \Big| = o\left( Ψ(x,y)^{1/2} \right),$$ 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 $x$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03430 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Helson's conjecture for smooth numbers Hardy, Seth Xu, Max Wenqiang Number Theory Classical Analysis and ODEs Complex Variables Functional Analysis Probability 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 always enjoy better than squareroot cancellation, in the sense that $$ \mathbb{E} \Big|\sum_{\substack{1\leq n \leq x\\ P(n) \leq y}} f(n) \Big| = o\left( Ψ(x,y)^{1/2} \right),$$ 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 $x$. |
| title | Helson's conjecture for smooth numbers |
| topic | Number Theory Classical Analysis and ODEs Complex Variables Functional Analysis Probability |
| url | https://arxiv.org/abs/2511.03430 |