Cusps of primes in dense subsequences -- Bypassing the $W$-trick
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913613772488704 |
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| author | Ramaré, Olivier |
| author_facet | Ramaré, Olivier |
| contents | Let the $A$-cusps of a dense subset $\mathcal{P}^*\in[\sqrt{N},N]$ of primes be points $α\in\mathbb{R}/\mathbb{Z}$ that are such that $|\sum_{\substack{p\in\mathcal{P}^*}} e(αp)|\ge |\mathcal{P}^*|/A$. We establish that any $(1/N)$-well spaced subset of $A$-cusps contains at most $20A^2K\log(2A)$ points, where $K=N/(|\mathcal{P}^*|\log N)$. We further show that any $B$-cusps~$ξ$ is accompanied, when $B\le \sqrt{A}$, by a large proportion of $A$-cusps of the shape $ξ+(a/q)$. We conclude this study by showing that, given $A\ge2$, the characteristic function $1_{\mathcal{P}^*}$ may be decomposed in the form $1_{\mathcal{P}^*}=(V(z_0)\log N)^{-1}f^\flat +f^\sharp$ where the trigonometric polynomial of $f^\sharp$ takes only values $\le |\mathcal{P}^*|/A$, and~$f^\flat$ is a bounded non-negative function supported on the integers prime to $M$; the parameters $z_0$ and $M$ are given in terms of~$A$, while $V(z_0)=\prod_{p<z_0}(1-1/p)$. The function $f^\flat$ satisfies more regularity properties. In particular, its density with respect to the integers $\le N$ and coprime to~$M$ is again~$K$. This transfers questions on~$\mathcal{P}^*$ to problems on integers coprime to the modulus~$M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11527 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cusps of primes in dense subsequences -- Bypassing the $W$-trick Ramaré, Olivier Number Theory Primary: 11N05, 11N36, Secondary: 11P32, 11L07, 11L20 Let the $A$-cusps of a dense subset $\mathcal{P}^*\in[\sqrt{N},N]$ of primes be points $α\in\mathbb{R}/\mathbb{Z}$ that are such that $|\sum_{\substack{p\in\mathcal{P}^*}} e(αp)|\ge |\mathcal{P}^*|/A$. We establish that any $(1/N)$-well spaced subset of $A$-cusps contains at most $20A^2K\log(2A)$ points, where $K=N/(|\mathcal{P}^*|\log N)$. We further show that any $B$-cusps~$ξ$ is accompanied, when $B\le \sqrt{A}$, by a large proportion of $A$-cusps of the shape $ξ+(a/q)$. We conclude this study by showing that, given $A\ge2$, the characteristic function $1_{\mathcal{P}^*}$ may be decomposed in the form $1_{\mathcal{P}^*}=(V(z_0)\log N)^{-1}f^\flat +f^\sharp$ where the trigonometric polynomial of $f^\sharp$ takes only values $\le |\mathcal{P}^*|/A$, and~$f^\flat$ is a bounded non-negative function supported on the integers prime to $M$; the parameters $z_0$ and $M$ are given in terms of~$A$, while $V(z_0)=\prod_{p<z_0}(1-1/p)$. The function $f^\flat$ satisfies more regularity properties. In particular, its density with respect to the integers $\le N$ and coprime to~$M$ is again~$K$. This transfers questions on~$\mathcal{P}^*$ to problems on integers coprime to the modulus~$M$. |
| title | Cusps of primes in dense subsequences -- Bypassing the $W$-trick |
| topic | Number Theory Primary: 11N05, 11N36, Secondary: 11P32, 11L07, 11L20 |
| url | https://arxiv.org/abs/2412.11527 |