Asymptotic independence of $Ω(n)$ and $Ω(n+1)$ along logarithmic averages
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| Format: | Preprint |
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2024
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| _version_ | 1866909498999832576 |
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| author | Charamaras, Dimitrios Richter, Florian K. |
| author_facet | Charamaras, Dimitrios Richter, Florian K. |
| contents | Let $Ω(n)$ denote the number of prime factors of a positive integer $n$ counted with multiplicities. We show that for any bounded functions $a,b\colon\mathbb{N}\to\mathbb{C}$, $$\frac{1}{\log{N}}\sum_{n=1}^N \frac{a(Ω(n))b(Ω(n+1))}{n} = \Bigg(\frac{1}{N}\sum_{n=1}^N a(Ω(n))\Bigg)\Bigg(\frac{1}{N}\sum_{n=1}^N b(Ω(n))\Bigg) + \mathrm{o}_{N\to\infty}(1).$$ This generalizes a theorem of Tao on the logarithmically averaged two-point correlation Chowla conjecture. Our result is quantitative and the explicit error term that we obtain establishes double-logarithmic savings. As an application, we obtain new results about the distribution of $Ω(p+1)$ as $p$ ranges over $\ell$-almost primes for a "typical" value of $\ell$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_17583 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic independence of $Ω(n)$ and $Ω(n+1)$ along logarithmic averages Charamaras, Dimitrios Richter, Florian K. Number Theory 11N37 (Primary) 11K65 (Secondary) Let $Ω(n)$ denote the number of prime factors of a positive integer $n$ counted with multiplicities. We show that for any bounded functions $a,b\colon\mathbb{N}\to\mathbb{C}$, $$\frac{1}{\log{N}}\sum_{n=1}^N \frac{a(Ω(n))b(Ω(n+1))}{n} = \Bigg(\frac{1}{N}\sum_{n=1}^N a(Ω(n))\Bigg)\Bigg(\frac{1}{N}\sum_{n=1}^N b(Ω(n))\Bigg) + \mathrm{o}_{N\to\infty}(1).$$ This generalizes a theorem of Tao on the logarithmically averaged two-point correlation Chowla conjecture. Our result is quantitative and the explicit error term that we obtain establishes double-logarithmic savings. As an application, we obtain new results about the distribution of $Ω(p+1)$ as $p$ ranges over $\ell$-almost primes for a "typical" value of $\ell$. |
| title | Asymptotic independence of $Ω(n)$ and $Ω(n+1)$ along logarithmic averages |
| topic | Number Theory 11N37 (Primary) 11K65 (Secondary) |
| url | https://arxiv.org/abs/2412.17583 |