Smooth integers and the Dickman $ρ$ function
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866909346942681088 |
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| author | Gorodetsky, Ofir |
| author_facet | Gorodetsky, Ofir |
| contents | We establish an asymptotic formula for $Ψ(x,y)$ whose shape is $x ρ(\log x/\log y)$ times correction factors. These factors take into account the contributions of zeta zeros and prime powers and the formula can be regarded as an (approximate) explicit formula for $Ψ(x,y)$. With this formula at hand we prove oscillation results for $Ψ(x,y)$, which resolve a question of Hildebrand on the range of validity of $Ψ(x,y) \asymp xρ(\log x/\log y)$. We also address a question of Pomerance on the range of validity of $Ψ(x,y) \ge x ρ(\log x/\log y)$.
Along the way we improve classical estimates for $Ψ(x,y)$ and, on the Riemann Hypothesis, uncover an unexpected phase transition of $Ψ(x,y)$ at $y=(\log x)^{3/2+o(1)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_08973 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Smooth integers and the Dickman $ρ$ function Gorodetsky, Ofir Number Theory We establish an asymptotic formula for $Ψ(x,y)$ whose shape is $x ρ(\log x/\log y)$ times correction factors. These factors take into account the contributions of zeta zeros and prime powers and the formula can be regarded as an (approximate) explicit formula for $Ψ(x,y)$. With this formula at hand we prove oscillation results for $Ψ(x,y)$, which resolve a question of Hildebrand on the range of validity of $Ψ(x,y) \asymp xρ(\log x/\log y)$. We also address a question of Pomerance on the range of validity of $Ψ(x,y) \ge x ρ(\log x/\log y)$. Along the way we improve classical estimates for $Ψ(x,y)$ and, on the Riemann Hypothesis, uncover an unexpected phase transition of $Ψ(x,y)$ at $y=(\log x)^{3/2+o(1)}$. |
| title | Smooth integers and the Dickman $ρ$ function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2211.08973 |