Diagonal cubic forms and the large sieve
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866929655962927104 |
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| author | Wang, Victor Y. |
| author_facet | Wang, Victor Y. |
| contents | Let $N(X)$ be the number of integral zeros $(x_1,\dots,x_6)\in [-X,X]^6$ of $\sum_{1\le i\le 6} x_i^3$. Works of Hooley and Heath-Brown imply $N(X)\ll_εX^{3+ε}$, if one assumes automorphy and GRH for certain Hasse--Weil $L$-functions. Assuming instead a natural large sieve inequality, we recover the same bound on $N(X)$. This is part of a more general statement, for diagonal cubic forms in $\geq 4$ variables, where we allow approximations to Hasse--Weil $L$-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_03395 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Diagonal cubic forms and the large sieve Wang, Victor Y. Number Theory Let $N(X)$ be the number of integral zeros $(x_1,\dots,x_6)\in [-X,X]^6$ of $\sum_{1\le i\le 6} x_i^3$. Works of Hooley and Heath-Brown imply $N(X)\ll_εX^{3+ε}$, if one assumes automorphy and GRH for certain Hasse--Weil $L$-functions. Assuming instead a natural large sieve inequality, we recover the same bound on $N(X)$. This is part of a more general statement, for diagonal cubic forms in $\geq 4$ variables, where we allow approximations to Hasse--Weil $L$-functions. |
| title | Diagonal cubic forms and the large sieve |
| topic | Number Theory |
| url | https://arxiv.org/abs/2108.03395 |