On the Density of naturals $n$ coprime to $\lfloor P(n) \rfloor$ for certain Classes of Polynomials
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913583191818240 |
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| author | Chatterjee, Aahan |
| author_facet | Chatterjee, Aahan |
| contents | We obtain asymptotic bounds on the number of natural numbers less than $X$ satisfying $\gcd{\left(n,\lfloor P(n) \rfloor\right)}=1$, under some diophantine conditions on the coefficient of $x$ in $P$, and show that the density of such naturals is exactly $\dfrac{1}{ζ{\left(2\right)}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11316 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Density of naturals $n$ coprime to $\lfloor P(n) \rfloor$ for certain Classes of Polynomials Chatterjee, Aahan Number Theory We obtain asymptotic bounds on the number of natural numbers less than $X$ satisfying $\gcd{\left(n,\lfloor P(n) \rfloor\right)}=1$, under some diophantine conditions on the coefficient of $x$ in $P$, and show that the density of such naturals is exactly $\dfrac{1}{ζ{\left(2\right)}}$. |
| title | On the Density of naturals $n$ coprime to $\lfloor P(n) \rfloor$ for certain Classes of Polynomials |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.11316 |