On the connection between zero-free regions and the error term in the Prime Number Theorem
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915398079741952 |
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| author | Broucke, Frederik |
| author_facet | Broucke, Frederik |
| contents | We provide for a wide class of zero-free regions an upper bound for the error term in the Prime Number Theorem, refining works of Pintz (1980), Johnston (2024), and Révész (2024). Our method does not only apply to the Riemann zeta function, but to general Beurling zeta functions. Next we construct Beurling zeta functions having infinitely many zeros on a prescribed contour, and none to the right, for a wide class of such contours. We also deduce an oscillation result for the corresponding error term in the Prime Number Theorem, showing that our aforementioned refinement is close to being sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13780 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the connection between zero-free regions and the error term in the Prime Number Theorem Broucke, Frederik Number Theory 11M26, 11N05, 11M41, 11N80 We provide for a wide class of zero-free regions an upper bound for the error term in the Prime Number Theorem, refining works of Pintz (1980), Johnston (2024), and Révész (2024). Our method does not only apply to the Riemann zeta function, but to general Beurling zeta functions. Next we construct Beurling zeta functions having infinitely many zeros on a prescribed contour, and none to the right, for a wide class of such contours. We also deduce an oscillation result for the corresponding error term in the Prime Number Theorem, showing that our aforementioned refinement is close to being sharp. |
| title | On the connection between zero-free regions and the error term in the Prime Number Theorem |
| topic | Number Theory 11M26, 11N05, 11M41, 11N80 |
| url | https://arxiv.org/abs/2507.13780 |