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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2408.16576 |
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| _version_ | 1866908705101971456 |
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| author | Benatar, Jacques |
| author_facet | Benatar, Jacques |
| contents | In the late eighties, Hildebrand and Tenenbaum proved an asymptotic formula for the number of positive integers below $x$, having exactly $ν$ distinct prime divisors: $π_ν(x) \sim x δ_ν(x)$. Here we consider the restricted count $π_ν(x,y)$ for integers lying in the short interval $(x,x+y]$. In this setting, we show that for any $\varepsilon >0$, the asymptotic equivalence \[ π_ν(x,y) \sim y δ_ν(x)\] holds uniformly over all $1 \le ν\le (\log x)^{1/3}/(\log \log x)^2$ and all $x^{17/30 + \varepsilon} \leq y \leq x$. The methods also furnish mean upper bounds for the $k$-fold divisor function $τ_k$ in short intervals, with strong uniformity in $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16576 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A short-interval Hildebrand-Tenenbaum theorem Benatar, Jacques Number Theory 11N25 In the late eighties, Hildebrand and Tenenbaum proved an asymptotic formula for the number of positive integers below $x$, having exactly $ν$ distinct prime divisors: $π_ν(x) \sim x δ_ν(x)$. Here we consider the restricted count $π_ν(x,y)$ for integers lying in the short interval $(x,x+y]$. In this setting, we show that for any $\varepsilon >0$, the asymptotic equivalence \[ π_ν(x,y) \sim y δ_ν(x)\] holds uniformly over all $1 \le ν\le (\log x)^{1/3}/(\log \log x)^2$ and all $x^{17/30 + \varepsilon} \leq y \leq x$. The methods also furnish mean upper bounds for the $k$-fold divisor function $τ_k$ in short intervals, with strong uniformity in $k$. |
| title | A short-interval Hildebrand-Tenenbaum theorem |
| topic | Number Theory 11N25 |
| url | https://arxiv.org/abs/2408.16576 |