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We prove under GRH explicit inequalities for the differences $|\\psi^{(1)}_{\\mathbb K}(x) - \\tfrac{x^2}{2}|$ and $|\\psi^{(2)}_{\\mathbb K}(x) - \\tfrac{x^3}{3}|$. We deduce an efficient algorithm for the computation of the residue of the Dedekind zeta function and a bound on small-norm prime ideals."},"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":"1312.4465","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-12-16T19:10:48Z","cross_cats_sorted":[],"title_canon_sha256":"8c0328a0d7e874136f10922b906e3282757b1469d3f3b368bdefb72a3dc9be13","abstract_canon_sha256":"e91adf10fd8d3f39fc52242fba66654db251c18b671a507a103288c9d5928e42"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:05.096635Z","signature_b64":"bx1eyaKOVs9B97rW900zlP3XSiK8H3s7jTixul+973LDGPG3xA8Rh+nyu6InMk/ZS/R1MtWe3vxiw2DoL3lCAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d4d8a1f7f29d9c57974d1c28ab2b94a6f39ce9bf0e67eb2c583396628d644e2","last_reissued_at":"2026-05-17T23:45:05.096107Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:05.096107Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Explicit smoothed prime ideals theorems under GRH","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Giuseppe Molteni, Lo\\\"ic Greni\\'e","submitted_at":"2013-12-16T19:10:48Z","abstract_excerpt":"Let $\\psi_{\\mathbb K}$ be the Chebyshev function of a number field $\\mathbb K$. Let $\\psi^{(1)}_{\\mathbb K}(x):=\\int_{0}^{x}\\psi_{\\mathbb K}(t)\\,d t$ and $\\psi^{(2)}_{\\mathbb K}(x):=2\\int_{0}^{x}\\psi^{(1)}_{\\mathbb K}(t)\\,d t$. We prove under GRH explicit inequalities for the differences $|\\psi^{(1)}_{\\mathbb K}(x) - \\tfrac{x^2}{2}|$ and $|\\psi^{(2)}_{\\mathbb K}(x) - \\tfrac{x^3}{3}|$. We deduce an efficient algorithm for the computation of the residue of the Dedekind zeta function and a bound on small-norm prime ideals."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.4465","kind":"arxiv","version":5},"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":"1312.4465","created_at":"2026-05-17T23:45:05.096188+00:00"},{"alias_kind":"arxiv_version","alias_value":"1312.4465v5","created_at":"2026-05-17T23:45:05.096188+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.4465","created_at":"2026-05-17T23:45:05.096188+00:00"},{"alias_kind":"pith_short_12","alias_value":"DVGYUH37FHM4","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_16","alias_value":"DVGYUH37FHM4K6LU","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_8","alias_value":"DVGYUH37","created_at":"2026-05-18T12:27:43.054852+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/DVGYUH37FHM4K6LU2HBIVMVZJJ","json":"https://pith.science/pith/DVGYUH37FHM4K6LU2HBIVMVZJJ.json","graph_json":"https://pith.science/api/pith-number/DVGYUH37FHM4K6LU2HBIVMVZJJ/graph.json","events_json":"https://pith.science/api/pith-number/DVGYUH37FHM4K6LU2HBIVMVZJJ/events.json","paper":"https://pith.science/paper/DVGYUH37"},"agent_actions":{"view_html":"https://pith.science/pith/DVGYUH37FHM4K6LU2HBIVMVZJJ","download_json":"https://pith.science/pith/DVGYUH37FHM4K6LU2HBIVMVZJJ.json","view_paper":"https://pith.science/paper/DVGYUH37","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1312.4465&json=true","fetch_graph":"https://pith.science/api/pith-number/DVGYUH37FHM4K6LU2HBIVMVZJJ/graph.json","fetch_events":"https://pith.science/api/pith-number/DVGYUH37FHM4K6LU2HBIVMVZJJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DVGYUH37FHM4K6LU2HBIVMVZJJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DVGYUH37FHM4K6LU2HBIVMVZJJ/action/storage_attestation","attest_author":"https://pith.science/pith/DVGYUH37FHM4K6LU2HBIVMVZJJ/action/author_attestation","sign_citation":"https://pith.science/pith/DVGYUH37FHM4K6LU2HBIVMVZJJ/action/citation_signature","submit_replication":"https://pith.science/pith/DVGYUH37FHM4K6LU2HBIVMVZJJ/action/replication_record"}},"created_at":"2026-05-17T23:45:05.096188+00:00","updated_at":"2026-05-17T23:45:05.096188+00:00"}