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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2503.05396 |
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| _version_ | 1866912264448114688 |
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| author | Merikoski, Jori |
| author_facet | Merikoski, Jori |
| contents | Let $a,b>0$ be coprime integers. Assuming a conjecture on Hecke eigenvalues along binary cubic forms, we prove an asymptotic formula for the number of primes of the form $ax^2+by^3$ with $x \leq X^{1/2}$ and $y \leq X^{1/3}$. The proof combines sieve methods with the theory of real quadratic fields/indefinite binary quadratic forms, the Weil bound for exponential sums, and spectral methods of GL(2) automorphic forms. We also discuss applications to elliptic curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05396 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On primes represented by $aX^2+bY^3$ Merikoski, Jori Number Theory Let $a,b>0$ be coprime integers. Assuming a conjecture on Hecke eigenvalues along binary cubic forms, we prove an asymptotic formula for the number of primes of the form $ax^2+by^3$ with $x \leq X^{1/2}$ and $y \leq X^{1/3}$. The proof combines sieve methods with the theory of real quadratic fields/indefinite binary quadratic forms, the Weil bound for exponential sums, and spectral methods of GL(2) automorphic forms. We also discuss applications to elliptic curves. |
| title | On primes represented by $aX^2+bY^3$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2503.05396 |