Asymptotics for smooth numbers in short intervals
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913494198124544 |
|---|---|
| author | Younis, Khalid |
| author_facet | Younis, Khalid |
| contents | A number is said to be $y$-smooth if all of its prime factors are less than or equal to $y.$ For all $17/30<θ\leq 1,$ we show that the density of $y$-smooth numbers in the short interval $[x,x+x^θ]$ is asymptotically equal to the density of $y$-smooth numbers in the long interval $[1,x],$ for all $y \geq \exp((\log x)^{2/3+\varepsilon}).$ Assuming the Riemann Hypothesis, we also prove that for all $1/2<θ\leq 1$ there exists a large constant $K$ such that the expected asymptotic result holds for $y\geq (\log x)^{K}.$
Our approach is to count smooth numbers using a Perron integral, shift this to a particular contour left of the saddle point, and employ a zero-density estimate of the Riemann zeta function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05761 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotics for smooth numbers in short intervals Younis, Khalid Number Theory A number is said to be $y$-smooth if all of its prime factors are less than or equal to $y.$ For all $17/30<θ\leq 1,$ we show that the density of $y$-smooth numbers in the short interval $[x,x+x^θ]$ is asymptotically equal to the density of $y$-smooth numbers in the long interval $[1,x],$ for all $y \geq \exp((\log x)^{2/3+\varepsilon}).$ Assuming the Riemann Hypothesis, we also prove that for all $1/2<θ\leq 1$ there exists a large constant $K$ such that the expected asymptotic result holds for $y\geq (\log x)^{K}.$ Our approach is to count smooth numbers using a Perron integral, shift this to a particular contour left of the saddle point, and employ a zero-density estimate of the Riemann zeta function. |
| title | Asymptotics for smooth numbers in short intervals |
| topic | Number Theory |
| url | https://arxiv.org/abs/2409.05761 |