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We study expressions of the form \\[\n  \\sum_{h \\bmod{a_k}} \\ \\prod_{\\substack{i=1\\\\ i\\not=k}}^{n} \\ \\B_{p_i}\\left(a_i \\frac{h+x_k}{a_k}-x_i\\right). \\] These \\highlight{Bernoulli--Dedekind sums} generalize and unify various arithmetic sums introduced by Dedekind, Apostol, Carlitz, Rademacher, Sczech, Hall--Wilson--Zagier, and others. Generalized Dedekind sums appear in various areas such as analytic and algebraic number theory, topology, algebraic and "},"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":"1008.0038","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-31T00:47:07Z","cross_cats_sorted":[],"title_canon_sha256":"9f70ec35d1726b6f799fde0a6c6ccaa02843e43a0e21839abd94c7bf26593640","abstract_canon_sha256":"518b0b449a8f65eb4787bdd2ffd5b027ddf6ca85526dae9428e5406b13be5374"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:11:24.324108Z","signature_b64":"Z6VH8vqE2QWbiZgDQVfAW+Botl3KkUadTXlWT3JCQnuEOybwybLGKfX8Wf/WovOWJkhX89eouSQYc7t7UyTlAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b96d5186845941f655790b8f9890dea5ba1453814c0fb44491a873c39cf7c9be","last_reissued_at":"2026-05-18T03:11:24.323443Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:11:24.323443Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bernoulli--Dedekind Sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Anastasia Chavez, Matthias Beck","submitted_at":"2010-07-31T00:47:07Z","abstract_excerpt":"Let $p_1,p_2,\\dots,p_n, a_1,a_2,\\dots,a_n \\in \\N$, $x_1,x_2,\\dots,x_n \\in \\R$, and denote the $k$th periodized Bernoulli polynomial by $\\B_k(x)$. We study expressions of the form \\[\n  \\sum_{h \\bmod{a_k}} \\ \\prod_{\\substack{i=1\\\\ i\\not=k}}^{n} \\ \\B_{p_i}\\left(a_i \\frac{h+x_k}{a_k}-x_i\\right). \\] These \\highlight{Bernoulli--Dedekind sums} generalize and unify various arithmetic sums introduced by Dedekind, Apostol, Carlitz, Rademacher, Sczech, Hall--Wilson--Zagier, and others. Generalized Dedekind sums appear in various areas such as analytic and algebraic number theory, topology, algebraic and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.0038","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1008.0038","created_at":"2026-05-18T03:11:24.323578+00:00"},{"alias_kind":"arxiv_version","alias_value":"1008.0038v1","created_at":"2026-05-18T03:11:24.323578+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.0038","created_at":"2026-05-18T03:11:24.323578+00:00"},{"alias_kind":"pith_short_12","alias_value":"XFWVDBUELFA7","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_16","alias_value":"XFWVDBUELFA7MVLZ","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_8","alias_value":"XFWVDBUE","created_at":"2026-05-18T12:26:17.028572+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XFWVDBUELFA7MVLZBOHZREG6UW","json":"https://pith.science/pith/XFWVDBUELFA7MVLZBOHZREG6UW.json","graph_json":"https://pith.science/api/pith-number/XFWVDBUELFA7MVLZBOHZREG6UW/graph.json","events_json":"https://pith.science/api/pith-number/XFWVDBUELFA7MVLZBOHZREG6UW/events.json","paper":"https://pith.science/paper/XFWVDBUE"},"agent_actions":{"view_html":"https://pith.science/pith/XFWVDBUELFA7MVLZBOHZREG6UW","download_json":"https://pith.science/pith/XFWVDBUELFA7MVLZBOHZREG6UW.json","view_paper":"https://pith.science/paper/XFWVDBUE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1008.0038&json=true","fetch_graph":"https://pith.science/api/pith-number/XFWVDBUELFA7MVLZBOHZREG6UW/graph.json","fetch_events":"https://pith.science/api/pith-number/XFWVDBUELFA7MVLZBOHZREG6UW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XFWVDBUELFA7MVLZBOHZREG6UW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XFWVDBUELFA7MVLZBOHZREG6UW/action/storage_attestation","attest_author":"https://pith.science/pith/XFWVDBUELFA7MVLZBOHZREG6UW/action/author_attestation","sign_citation":"https://pith.science/pith/XFWVDBUELFA7MVLZBOHZREG6UW/action/citation_signature","submit_replication":"https://pith.science/pith/XFWVDBUELFA7MVLZBOHZREG6UW/action/replication_record"}},"created_at":"2026-05-18T03:11:24.323578+00:00","updated_at":"2026-05-18T03:11:24.323578+00:00"}