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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.09021 |
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| _version_ | 1866913117512925184 |
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| author | Bloom, Thomas F. Kuperberg, Vivian |
| author_facet | Bloom, Thomas F. Kuperberg, Vivian |
| contents | We prove near-optimal upper bounds for the odd moments of the distribution of coprime residues in short intervals, confirming a conjecture of Montgomery and Vaughan. As an application we prove near-optimal upper bounds for the average of the refined singular series in the Hardy-Littlewood conjectures concerning the number of prime $k$-tuples for $k$ odd. The main new ingredient is a near-optimal upper bound for the number of solutions to $\sum_{1\leq i\leq k}\frac{a_i}{q_i}\in \mathbb{Z}$ when $k$ is odd, with $(a_i,q_i)=1$ and restrictions on the size of the numerators and denominators, that is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_09021 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Odd moments and adding fractions Bloom, Thomas F. Kuperberg, Vivian Number Theory We prove near-optimal upper bounds for the odd moments of the distribution of coprime residues in short intervals, confirming a conjecture of Montgomery and Vaughan. As an application we prove near-optimal upper bounds for the average of the refined singular series in the Hardy-Littlewood conjectures concerning the number of prime $k$-tuples for $k$ odd. The main new ingredient is a near-optimal upper bound for the number of solutions to $\sum_{1\leq i\leq k}\frac{a_i}{q_i}\in \mathbb{Z}$ when $k$ is odd, with $(a_i,q_i)=1$ and restrictions on the size of the numerators and denominators, that is of independent interest. |
| title | Odd moments and adding fractions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2312.09021 |