On the series expansion of the prime zeta function about $s=1$ and its coefficients
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866912978380521472 |
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| author | Kawalec, Artur |
| author_facet | Kawalec, Artur |
| contents | In this article, we derive a series expansion of the prime zeta function about the $s=1$ logarithmic singularity and prove general formula for its expansion coefficients, which is similar to the Stieltjes expansion coefficients for the Riemann zeta function. These results can also be viewed as a generalization of Mertens's Theorems to higher order. We also numerically verify and compute the presented formulas to high precision for several test cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21535 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the series expansion of the prime zeta function about $s=1$ and its coefficients Kawalec, Artur Number Theory In this article, we derive a series expansion of the prime zeta function about the $s=1$ logarithmic singularity and prove general formula for its expansion coefficients, which is similar to the Stieltjes expansion coefficients for the Riemann zeta function. These results can also be viewed as a generalization of Mertens's Theorems to higher order. We also numerically verify and compute the presented formulas to high precision for several test cases. |
| title | On the series expansion of the prime zeta function about $s=1$ and its coefficients |
| topic | Number Theory |
| url | https://arxiv.org/abs/2603.21535 |