Local divisor correlations in almost all short intervals
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912259950772224 |
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| author | Pliego, Javier Sun, Yu-Chen Wang, Mengdi |
| author_facet | Pliego, Javier Sun, Yu-Chen Wang, Mengdi |
| contents | Let $ k,l \geq 2$ be natural numbers, and let $d_k,d_l$ denote the $k$-fold and $l$-fold divisor functions, respectively. We analyse the asymptotic behavior of the sum $\sum_{x<n\leq x+H_1}d_k(n)d_l(n+h)$. More precisely, let $\varepsilon>0$ be a small fixed number and let $Φ(x)$ be a positive function that tends to infinity arbitrarily slowly as $x\to \infty$. We then show that whenever $H_1\geq(\log x)^{Φ(x)}$ and $(\log x)^{1000k\log k}\leq H_2\leq H_1^{1-\varepsilon }$, the expected asymptotic formula holds for almost all $x\in[X,2X]$ and almost all $1\leq h\leq H_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02962 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local divisor correlations in almost all short intervals Pliego, Javier Sun, Yu-Chen Wang, Mengdi Number Theory Let $ k,l \geq 2$ be natural numbers, and let $d_k,d_l$ denote the $k$-fold and $l$-fold divisor functions, respectively. We analyse the asymptotic behavior of the sum $\sum_{x<n\leq x+H_1}d_k(n)d_l(n+h)$. More precisely, let $\varepsilon>0$ be a small fixed number and let $Φ(x)$ be a positive function that tends to infinity arbitrarily slowly as $x\to \infty$. We then show that whenever $H_1\geq(\log x)^{Φ(x)}$ and $(\log x)^{1000k\log k}\leq H_2\leq H_1^{1-\varepsilon }$, the expected asymptotic formula holds for almost all $x\in[X,2X]$ and almost all $1\leq h\leq H_2$. |
| title | Local divisor correlations in almost all short intervals |
| topic | Number Theory |
| url | https://arxiv.org/abs/2503.02962 |