On the representation of $k$-free integers by binary forms
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arXiv
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| Format: | Preprint |
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2016
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| _version_ | 1866929409374552064 |
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| author | Stewart, C. L. Xiao, Stanley Yao |
| author_facet | Stewart, C. L. Xiao, Stanley Yao |
| contents | Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d$ with $d$ at least $3$ and let $r$ denote the largest degree of an irreducible factor of $F$ over the rationals. Let $k$ be an integer with $k \geq 2$ and suppose that there is no prime $p$ such that $p^k$ divides $F(a,b)$ for all pairs of integers $(a,b)$. Let $R_{F,k}(Z)$ denote the number of $k$-free integers of absolute value at most $Z$ which are represented by $F$. We prove that there is a positive number $C_{F,k}$ such that $R_{F,k}(Z)$ is asymptotic to $C_{F,k} Z^{\frac{2}{d}}$ provided that $k$ exceeds $ \frac{7r}{18}$ or $(k,r)$ is $(2,6)$ or $(3,8)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1612_00487 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | On the representation of $k$-free integers by binary forms Stewart, C. L. Xiao, Stanley Yao Number Theory 11N32, 11E76 Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d$ with $d$ at least $3$ and let $r$ denote the largest degree of an irreducible factor of $F$ over the rationals. Let $k$ be an integer with $k \geq 2$ and suppose that there is no prime $p$ such that $p^k$ divides $F(a,b)$ for all pairs of integers $(a,b)$. Let $R_{F,k}(Z)$ denote the number of $k$-free integers of absolute value at most $Z$ which are represented by $F$. We prove that there is a positive number $C_{F,k}$ such that $R_{F,k}(Z)$ is asymptotic to $C_{F,k} Z^{\frac{2}{d}}$ provided that $k$ exceeds $ \frac{7r}{18}$ or $(k,r)$ is $(2,6)$ or $(3,8)$. |
| title | On the representation of $k$-free integers by binary forms |
| topic | Number Theory 11N32, 11E76 |
| url | https://arxiv.org/abs/1612.00487 |