On the series expansion of k-free Dirichlet series and its analytical continuation
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866918406174801920 |
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| author | Kawalec, Artur |
| author_facet | Kawalec, Artur |
| contents | In this article, we develop a k-free zeta Dirichlet series into a Laurent series with a simple pole, and prove a Stieltjes like formula for the expansion coefficients of the regular part. We also investigate another analytical continuation of these series and develop a formula for $ζ(\tfrac{1}{k})$ for positive integer $k\geq 2$ in terms of the k-free indicator function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22775 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the series expansion of k-free Dirichlet series and its analytical continuation Kawalec, Artur Number Theory In this article, we develop a k-free zeta Dirichlet series into a Laurent series with a simple pole, and prove a Stieltjes like formula for the expansion coefficients of the regular part. We also investigate another analytical continuation of these series and develop a formula for $ζ(\tfrac{1}{k})$ for positive integer $k\geq 2$ in terms of the k-free indicator function. |
| title | On the series expansion of k-free Dirichlet series and its analytical continuation |
| topic | Number Theory |
| url | https://arxiv.org/abs/2603.22775 |