Sharper bounds for the error in the prime number theorem assuming the Riemann Hypothesis
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914070860398592 |
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| author | Lee, Ethan Simpson Nosal, Paweł |
| author_facet | Lee, Ethan Simpson Nosal, Paweł |
| contents | In this paper, we establish new bounds for classical prime-counting functions. All of our bounds are explicit and assume the Riemann Hypothesis. First, we prove that $|ψ(x) - x|$ and $|\vartheta(x) - x|$ are bounded from above by $$\frac{\sqrt{x}\log{x}(\log{x} - \log\log{x})}{8π}$$ for all $x\geq 101$ and $x \geq 2\,657$ respectively, where $ψ(x)$ and $\vartheta(x)$ are the Chebyshev $ψ$ and $\vartheta$ functions. Using the extra precision offered by these results, we also prove new explicit descriptions for the error in each of Mertens' theorems which improve earlier bounds by Schoenfeld. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_05628 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sharper bounds for the error in the prime number theorem assuming the Riemann Hypothesis Lee, Ethan Simpson Nosal, Paweł Number Theory 11N05, 11L20, 11Y60, 11Y60, 11Y35 In this paper, we establish new bounds for classical prime-counting functions. All of our bounds are explicit and assume the Riemann Hypothesis. First, we prove that $|ψ(x) - x|$ and $|\vartheta(x) - x|$ are bounded from above by $$\frac{\sqrt{x}\log{x}(\log{x} - \log\log{x})}{8π}$$ for all $x\geq 101$ and $x \geq 2\,657$ respectively, where $ψ(x)$ and $\vartheta(x)$ are the Chebyshev $ψ$ and $\vartheta$ functions. Using the extra precision offered by these results, we also prove new explicit descriptions for the error in each of Mertens' theorems which improve earlier bounds by Schoenfeld. |
| title | Sharper bounds for the error in the prime number theorem assuming the Riemann Hypothesis |
| topic | Number Theory 11N05, 11L20, 11Y60, 11Y60, 11Y35 |
| url | https://arxiv.org/abs/2312.05628 |