{"paper":{"title":"An exotic shuffle relation of $\\zeta(\\{2\\}^m)$ and $\\zeta(\\{3,1\\}^n)$","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Jianqiang Zhao","submitted_at":"2007-07-23T10:34:21Z","abstract_excerpt":"In this short note we will provide a new and shorter proof of the following exotic shuffle relation of multiple zeta values:\n  $$\\zeta(\\{2\\}^m \\sha\\{3,1\\}^n)={2n+m\\choose m}\n  \\frac{\\pi^{4n+2m}}{(2n+1)\\cdot (4n+2m+1)!}.$$ This was proved by Zagier when n=0, by Broadhurst when $m=0$, and by Borwein, Bradley, and Broadhurst when m=1. In general this was proved by Bowman and Bradley in \\emph{The algebra and combinatorics of shuffles and multiple zeta values}, J. of Combinatorial Theory, Series A, Vol. \\textbf{97} (1)(2002), 43--63. Our idea in the general case is to use the method of Borwein et a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0707.3244","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}