Fast converging irrational series for $ L(2,(\frac d\cdot))$

Zhi-Wei Sun; Yajun Zhou

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arXiv · 2506.01865

Fast converging irrational series for $ L(2,(\frac d\cdot))$

Zhi-Wei Sun·Yajun Zhou

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Abstract

By exploring the theory of Guillera-Rogers, we evaluate some infinite series whose summands are quadratic irrationals, in terms of $\pi$ 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.

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BibTeX RIS

Zhi-Wei Sun, Yajun Zhou. 2025-06-02. Fast converging irrational series for $ L(2,(\frac d\cdot))$. https://arxiv.org/abs/2506.01865

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