Smooth Numbers in Short Intervals
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866916614679560192 |
|---|---|
| author | Jain, Sarvagya |
| author_facet | Jain, Sarvagya |
| contents | Let \( X \geq y \geq 2 \), and let \( u = \frac{\log X}{\log y} \). We say a number is \textit{$y$-smooth} if all of its prime factors are less than or equal to \( y \). In this paper, we study the distribution of $y$-smooth numbers in short intervals. In particular, for \( y \geq \exp\left( (\log X)^{2/3 + ε} \right) \), we show that the interval \( [x, x+h] \) contains a $y$-smooth number for almost all \( x \in [X, 2X] \), provided \( h \geq \exp\left( (1 + ε) \left( \frac{11}{8} u \log u + 4 \log \log X \right) \right) \), and \( X \) is sufficiently large depending on \( ε\). This result improves upon an earlier result by Matomäki. Additionally, we provide the corresponding ``all intervals" type result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10530 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Smooth Numbers in Short Intervals Jain, Sarvagya Number Theory 11N25 Let \( X \geq y \geq 2 \), and let \( u = \frac{\log X}{\log y} \). We say a number is \textit{$y$-smooth} if all of its prime factors are less than or equal to \( y \). In this paper, we study the distribution of $y$-smooth numbers in short intervals. In particular, for \( y \geq \exp\left( (\log X)^{2/3 + ε} \right) \), we show that the interval \( [x, x+h] \) contains a $y$-smooth number for almost all \( x \in [X, 2X] \), provided \( h \geq \exp\left( (1 + ε) \left( \frac{11}{8} u \log u + 4 \log \log X \right) \right) \), and \( X \) is sufficiently large depending on \( ε\). This result improves upon an earlier result by Matomäki. Additionally, we provide the corresponding ``all intervals" type result. |
| title | Smooth Numbers in Short Intervals |
| topic | Number Theory 11N25 |
| url | https://arxiv.org/abs/2502.10530 |