{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:VJXPMSIAZQHNPUJL7GIOBZ5LHA","short_pith_number":"pith:VJXPMSIA","schema_version":"1.0","canonical_sha256":"aa6ef64900cc0ed7d12bf990e0e7ab381e36a5320c710ef0beb1998db6e801fc","source":{"kind":"arxiv","id":"1111.1950","version":1},"attestation_state":"computed","paper":{"title":"Exact & Numerical Tests of Generalised Root Identities for non-integer \\mu","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Richard Stone","submitted_at":"2011-11-07T12:09:50Z","abstract_excerpt":"We consider the generalised root identities introduced in [1] for simple functions, and also for \\Gamma(z+1) and \\zeta(s). In this paper, unlike [1], we focus on the case of noninteger \\mu. For the simplest function f(z)=z, and hence for arbitrary polynomials, we show that they are satisfied for arbitrary real {\\mu} (and hence for arbitrary complex {\\mu} by analytic continuation). Using this, we then develop an asymptotic formula for the derivative side of the root identities for \\Gamma(z+1) at arbitrary real \\mu, from which we are able to demonstrate numerically that \\Gamma(z+1) also satisfie"},"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":"1111.1950","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-11-07T12:09:50Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"ba54bd90316bc81132c5880b74c68e3c1524249aa9079a5bdb442a862b269215","abstract_canon_sha256":"ccaac11a598dec6218fac7088901f1513aea4f42bab9a21e5940912e22232f08"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:24.330619Z","signature_b64":"Y6HvWtRSiRIvvHjEzfgv30bxZri5JAaPP/9v9obWHNOVe4xRRbgkdv8u+tgHckgM+BI9zshZ8NyO2a4DRHxdBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa6ef64900cc0ed7d12bf990e0e7ab381e36a5320c710ef0beb1998db6e801fc","last_reissued_at":"2026-05-18T04:09:24.329898Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:24.329898Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exact & Numerical Tests of Generalised Root Identities for non-integer \\mu","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Richard Stone","submitted_at":"2011-11-07T12:09:50Z","abstract_excerpt":"We consider the generalised root identities introduced in [1] for simple functions, and also for \\Gamma(z+1) and \\zeta(s). In this paper, unlike [1], we focus on the case of noninteger \\mu. For the simplest function f(z)=z, and hence for arbitrary polynomials, we show that they are satisfied for arbitrary real {\\mu} (and hence for arbitrary complex {\\mu} by analytic continuation). Using this, we then develop an asymptotic formula for the derivative side of the root identities for \\Gamma(z+1) at arbitrary real \\mu, from which we are able to demonstrate numerically that \\Gamma(z+1) also satisfie"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.1950","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":"1111.1950","created_at":"2026-05-18T04:09:24.330007+00:00"},{"alias_kind":"arxiv_version","alias_value":"1111.1950v1","created_at":"2026-05-18T04:09:24.330007+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.1950","created_at":"2026-05-18T04:09:24.330007+00:00"},{"alias_kind":"pith_short_12","alias_value":"VJXPMSIAZQHN","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_16","alias_value":"VJXPMSIAZQHNPUJL","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_8","alias_value":"VJXPMSIA","created_at":"2026-05-18T12:26:44.992195+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/VJXPMSIAZQHNPUJL7GIOBZ5LHA","json":"https://pith.science/pith/VJXPMSIAZQHNPUJL7GIOBZ5LHA.json","graph_json":"https://pith.science/api/pith-number/VJXPMSIAZQHNPUJL7GIOBZ5LHA/graph.json","events_json":"https://pith.science/api/pith-number/VJXPMSIAZQHNPUJL7GIOBZ5LHA/events.json","paper":"https://pith.science/paper/VJXPMSIA"},"agent_actions":{"view_html":"https://pith.science/pith/VJXPMSIAZQHNPUJL7GIOBZ5LHA","download_json":"https://pith.science/pith/VJXPMSIAZQHNPUJL7GIOBZ5LHA.json","view_paper":"https://pith.science/paper/VJXPMSIA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1111.1950&json=true","fetch_graph":"https://pith.science/api/pith-number/VJXPMSIAZQHNPUJL7GIOBZ5LHA/graph.json","fetch_events":"https://pith.science/api/pith-number/VJXPMSIAZQHNPUJL7GIOBZ5LHA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VJXPMSIAZQHNPUJL7GIOBZ5LHA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VJXPMSIAZQHNPUJL7GIOBZ5LHA/action/storage_attestation","attest_author":"https://pith.science/pith/VJXPMSIAZQHNPUJL7GIOBZ5LHA/action/author_attestation","sign_citation":"https://pith.science/pith/VJXPMSIAZQHNPUJL7GIOBZ5LHA/action/citation_signature","submit_replication":"https://pith.science/pith/VJXPMSIAZQHNPUJL7GIOBZ5LHA/action/replication_record"}},"created_at":"2026-05-18T04:09:24.330007+00:00","updated_at":"2026-05-18T04:09:24.330007+00:00"}