{"paper":{"title":"Fast converging irrational series for $ L(2,(\\frac d\\cdot))$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Yajun Zhou, Zhi-Wei Sun","submitted_at":"2025-06-02T16:53:21Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.01865","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.01865/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}