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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2504.16002 |
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| _version_ | 1866909791511642112 |
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| author | Wang, Biao |
| author_facet | Wang, Biao |
| contents | Let $μ(n)$ be the Möbius function. Let $P^-(n)$ denote the smallest prime factor of an integer $n$. In 1977, Alladi established the following formula related to the prime number theorem for arithmetic progressions \[
-\sum_{\substack{n\geq 2\\ P^-(n)\equiv \ell ({\rm mod}k)}}\frac{μ(n)}{n}=\frac1{φ(k)} \] for positive integers $\ell, k\ge$ with $(\ell,k)=1$, where $φ$ is Euler's totient function. In this note, we will show a logarithmic analogue of Alladi's formula in an elementary proof. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16002 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A logarithmic analogue of Alladi's formula Wang, Biao Number Theory Let $μ(n)$ be the Möbius function. Let $P^-(n)$ denote the smallest prime factor of an integer $n$. In 1977, Alladi established the following formula related to the prime number theorem for arithmetic progressions \[ -\sum_{\substack{n\geq 2\\ P^-(n)\equiv \ell ({\rm mod}k)}}\frac{μ(n)}{n}=\frac1{φ(k)} \] for positive integers $\ell, k\ge$ with $(\ell,k)=1$, where $φ$ is Euler's totient function. In this note, we will show a logarithmic analogue of Alladi's formula in an elementary proof. |
| title | A logarithmic analogue of Alladi's formula |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.16002 |