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For example, we show that for any prime $p>3$ we have $$\\sum_{k=1}^{p-1}\\frac{H_k}{k2^k}\\equiv\\frac7{24}pB_{p-3}\\pmod{p^2},\\ \\ \\sum_{k=1}^{p-1}\\frac{H_{k,2}}{k2^k}\\equiv-\\frac 38B_{p-3}\\pmod{p},$$ and $$\\sum_{k=1}^{p-1}\\frac{H_{k,2n}^2}{k^{2n}}\\equiv\\frac{\\binom{6n+1}{2n-1}+n}{6n+1}pB_{p-1-6n}\\pmod{p^2}$$ for any positive integer $n<(p-1)/6$, where $B_0,B_1,B_2,\\ldots$ are Bernoulli numbers, and $H_{k,m}:=\\sum_{j=1}^k 1/j^m$."},"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":"0911.4433","kind":"arxiv","version":8},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-11-23T19:27:44Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a58cad07c8e490482620d02c9ad1881e9893c098b66878e13c052246a9a702fa","abstract_canon_sha256":"efe5d1e0895e01f91a58c317268b29ac2571e6f58ba5d02e9c36ecd20d480d71"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:08:27.557868Z","signature_b64":"BGiOMbdabxuSpz5g0T21l7lgOTsLsac3jYZVxgeSpahLTWpEc/ZylZQkLMHnsFqpp3yT3Ha6HtwQPvNqGcalCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"214e0dc8e498dbb9086d960673018e94cf74048c8070e57552d8b5aafa4e06eb","last_reissued_at":"2026-05-18T03:08:27.556994Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:08:27.556994Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Arithmetic theory of harmonic numbers (II)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Li-Lu Zhao, Zhi-Wei Sun","submitted_at":"2009-11-23T19:27:44Z","abstract_excerpt":"For $k=1,2,\\ldots$ let $H_k$ denote the harmonic number $\\sum_{j=1}^k 1/j$. 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