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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0707.3244","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2007-07-23T10:34:21Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"384602d15c360254ec887e191c503234e42e6730cf08543197cb916a0ab61628","abstract_canon_sha256":"b823fd762a588894287ed7d71e1ebec41ce46348effec08edea0b027ac8f0c5e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:00.685508Z","signature_b64":"5PJpzr0nVvPK8gspQC7Z1IMEw2yy9NjGT4md8L7pqi76K8EaJkr7/mmv+UyqQI3xbun6x2Fxl+cm6IO4iDXGAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ca890824b77375fd4b01d7ce45f64c6c98017e70977a3c4f7bc9ba93cab0b5e3","last_reissued_at":"2026-05-18T04:26:00.685112Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:00.685112Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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