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In this paper we establish some new congruences involving harmonic numbers. 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 $H_{k,m}:=\\sum_{j=1}^k 1/j^m$.","authors_text":"Li-Lu Zhao, Zhi-Wei Sun","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-11-23T19:27:44Z","title":"Arithmetic theory of harmonic numbers (II)"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.4433","kind":"arxiv","version":8},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:efcfeb6df791ee001847951088c4511f641af08aa6366375900aefa3d568cdc7","target":"record","created_at":"2026-05-18T03:08:27Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"efe5d1e0895e01f91a58c317268b29ac2571e6f58ba5d02e9c36ecd20d480d71","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-11-23T19:27:44Z","title_canon_sha256":"a58cad07c8e490482620d02c9ad1881e9893c098b66878e13c052246a9a702fa"},"schema_version":"1.0","source":{"id":"0911.4433","kind":"arxiv","version":8}},"canonical_sha256":"214e0dc8e498dbb9086d960673018e94cf74048c8070e57552d8b5aafa4e06eb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"214e0dc8e498dbb9086d960673018e94cf74048c8070e57552d8b5aafa4e06eb","first_computed_at":"2026-05-18T03:08:27.556994Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:08:27.556994Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BGiOMbdabxuSpz5g0T21l7lgOTsLsac3jYZVxgeSpahLTWpEc/ZylZQkLMHnsFqpp3yT3Ha6HtwQPvNqGcalCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:08:27.557868Z","signed_message":"canonical_sha256_bytes"},"source_id":"0911.4433","source_kind":"arxiv","source_version":8}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:efcfeb6df791ee001847951088c4511f641af08aa6366375900aefa3d568cdc7","sha256:e8d8d1530eb95dd868357f121cbbcde6116bf5d92784d2b39393915e22823799"],"state_sha256":"74519d181f3914d542d773799f106dee96d0db4e8005c79d469a791f6131d0e6"}