Fast converging irrational series for $ L(2,(\frac d\cdot))$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915387711422464 |
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| author | Sun, Zhi-Wei Zhou, Yajun |
| author_facet | Sun, Zhi-Wei Zhou, Yajun |
| contents | By exploring the theory of Guillera-Rogers, we evaluate some infinite series whose summands are quadratic irrationals, in terms of $π$ and special values of Dirichlet $L$-functions $ L_d(2)\equiv L(2,(\frac d\cdot)):=\sum_{k=1}^\infty\left( \frac{d}{k} \right)\frac1{k^2}$. Applying Kronecker's theorem to linear combinations of lattice sums, we obtain geometrically convergent series for $ L_{-56}(2)$, $ L_{-68}(2)$, $ L_{-87}(2)$, $ L_{-111}(2)$, and $ L_{-116}(2)$, which go beyond the solvable cases of Guillera-Rogers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01865 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fast converging irrational series for $ L(2,(\frac d\cdot))$ Sun, Zhi-Wei Zhou, Yajun Number Theory Primary 11M06, 11F03, 11B65, Secondary 05A19, 33F05 By exploring the theory of Guillera-Rogers, we evaluate some infinite series whose summands are quadratic irrationals, in terms of $π$ and special values of Dirichlet $L$-functions $ L_d(2)\equiv L(2,(\frac d\cdot)):=\sum_{k=1}^\infty\left( \frac{d}{k} \right)\frac1{k^2}$. Applying Kronecker's theorem to linear combinations of lattice sums, we obtain geometrically convergent series for $ L_{-56}(2)$, $ L_{-68}(2)$, $ L_{-87}(2)$, $ L_{-111}(2)$, and $ L_{-116}(2)$, which go beyond the solvable cases of Guillera-Rogers. |
| title | Fast converging irrational series for $ L(2,(\frac d\cdot))$ |
| topic | Number Theory Primary 11M06, 11F03, 11B65, Secondary 05A19, 33F05 |
| url | https://arxiv.org/abs/2506.01865 |