The $S$-$E$ route to the Chebyshev bounds for the prime-counting function
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911737871073280 |
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| author | Hubbard, Kai |
| author_facet | Hubbard, Kai |
| contents | The Chebyshev bounds for the prime-counting function, i.e., $π(x) \asymp x/\ln x$, is established in a new way. This new approach follows the outline $π(x) \asymp \bigl(\sum_{p \le x} \sqrt{\ln p / p}\bigr)^{2} \asymp x/\ln x$. Here, the second $\asymp$ is derived from the classical estimate by Mertens, i.e., $\sum_{p \le x} (\ln p)/p = \ln x + O(1)$; while the first $\asymp$ is proved by considering the difference $\bigl(\sum_{p \le x} \sqrt{\ln p/p}\bigr)^{2} - \sum_{p \le x} (\ln p)/p$, which is shown as having the same order as $π(x)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21946 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $S$-$E$ route to the Chebyshev bounds for the prime-counting function Hubbard, Kai General Mathematics 11N05 (Primary), 11A41 (Secondary) The Chebyshev bounds for the prime-counting function, i.e., $π(x) \asymp x/\ln x$, is established in a new way. This new approach follows the outline $π(x) \asymp \bigl(\sum_{p \le x} \sqrt{\ln p / p}\bigr)^{2} \asymp x/\ln x$. Here, the second $\asymp$ is derived from the classical estimate by Mertens, i.e., $\sum_{p \le x} (\ln p)/p = \ln x + O(1)$; while the first $\asymp$ is proved by considering the difference $\bigl(\sum_{p \le x} \sqrt{\ln p/p}\bigr)^{2} - \sum_{p \le x} (\ln p)/p$, which is shown as having the same order as $π(x)$. |
| title | The $S$-$E$ route to the Chebyshev bounds for the prime-counting function |
| topic | General Mathematics 11N05 (Primary), 11A41 (Secondary) |
| url | https://arxiv.org/abs/2604.21946 |