Smooth integers and de Bruijn's approximation $Λ$
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916225227948032 |
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| author | Gorodetsky, Ofir |
| author_facet | Gorodetsky, Ofir |
| contents | This paper is concerned with the relationship of $y$-smooth integers and de Bruijn's approximation $Λ(x,y)$. Under the Riemann hypothesis, Saias proved that the count of $y$-smooth integers up to $x$, $Ψ(x,y)$, is asymptotic to $Λ(x,y)$ when $y \ge (\log x)^{2+\varepsilon}$. We extend the range to $y \ge (\log x)^{3/2+\varepsilon}$ by introducing a correction factor that takes into account the contributions of zeta zeros and prime powers.
We use this correction term to uncover a lower order term in the asymptotics of $Ψ(x,y)/Λ(x,y)$. The term relates to the error term in the prime number theorem, and implies that large positive (resp. negative) values of $\sum_{n \le y} Λ(n)-y$ lead to large positive (resp. negative) values of $Ψ(x,y)-Λ(x,y)$, and vice versa. Under the Linear Independence hypothesis, we show a Chebyshev's bias in $Ψ(x,y)-Λ(x,y)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_01949 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Smooth integers and de Bruijn's approximation $Λ$ Gorodetsky, Ofir Number Theory This paper is concerned with the relationship of $y$-smooth integers and de Bruijn's approximation $Λ(x,y)$. Under the Riemann hypothesis, Saias proved that the count of $y$-smooth integers up to $x$, $Ψ(x,y)$, is asymptotic to $Λ(x,y)$ when $y \ge (\log x)^{2+\varepsilon}$. We extend the range to $y \ge (\log x)^{3/2+\varepsilon}$ by introducing a correction factor that takes into account the contributions of zeta zeros and prime powers. We use this correction term to uncover a lower order term in the asymptotics of $Ψ(x,y)/Λ(x,y)$. The term relates to the error term in the prime number theorem, and implies that large positive (resp. negative) values of $\sum_{n \le y} Λ(n)-y$ lead to large positive (resp. negative) values of $Ψ(x,y)-Λ(x,y)$, and vice versa. Under the Linear Independence hypothesis, we show a Chebyshev's bias in $Ψ(x,y)-Λ(x,y)$. |
| title | Smooth integers and de Bruijn's approximation $Λ$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2212.01949 |