{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:42EHHOPIEX6FRKXOFDVRCLBOQG","short_pith_number":"pith:42EHHOPI","schema_version":"1.0","canonical_sha256":"e68873b9e825fc58aaee28eb112c2e8198a09f5861ed111e3367e9ae1dcdfb0c","source":{"kind":"arxiv","id":"1409.6280","version":2},"attestation_state":"computed","paper":{"title":"An identity connecting theta series associated with binary quadratic forms of discriminant $\\Delta$ and $\\Delta($prime$)^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Frank Patane","submitted_at":"2014-09-22T18:57:24Z","abstract_excerpt":"We state and prove an identity which connects theta series associated with binary quadratic forms of idoneal discriminants $\\Delta$ and $\\Delta p^2$, for $p$ a prime. Employing this identity, we extend the results of Toh by writing the theta series of forms of discriminant $\\Delta p^2$ as a linear combination of Lambert series. We then use these Lambert series decompositions to give explicit representation formulas for the forms of discriminant $\\Delta p^2$. Lastly, we give a generalization of our main identity, which employs a map of Buell to connect forms of discriminant $\\Delta$ to $\\Delta "},"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":"1409.6280","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-22T18:57:24Z","cross_cats_sorted":[],"title_canon_sha256":"b11274f5da0a531ea62fcec4eacc96b6571d4b98c41492e076339b98b816935c","abstract_canon_sha256":"def15b385430da86f6e7fca67b8c01a4554f31d6a658d69dcbe18b632cb9998e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:40:46.937400Z","signature_b64":"tGCcZXYqZLBJ+e7GxCWHu3JtFR/9JwI3UvxoH/WLq1r0FB7ZdP/VVa5XgaMTCfBqzBZWzb+U35eIQtoKS07IAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e68873b9e825fc58aaee28eb112c2e8198a09f5861ed111e3367e9ae1dcdfb0c","last_reissued_at":"2026-05-18T02:40:46.936988Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:40:46.936988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An identity connecting theta series associated with binary quadratic forms of discriminant $\\Delta$ and $\\Delta($prime$)^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Frank Patane","submitted_at":"2014-09-22T18:57:24Z","abstract_excerpt":"We state and prove an identity which connects theta series associated with binary quadratic forms of idoneal discriminants $\\Delta$ and $\\Delta p^2$, for $p$ a prime. Employing this identity, we extend the results of Toh by writing the theta series of forms of discriminant $\\Delta p^2$ as a linear combination of Lambert series. We then use these Lambert series decompositions to give explicit representation formulas for the forms of discriminant $\\Delta p^2$. Lastly, we give a generalization of our main identity, which employs a map of Buell to connect forms of discriminant $\\Delta$ to $\\Delta "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.6280","kind":"arxiv","version":2},"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":"1409.6280","created_at":"2026-05-18T02:40:46.937045+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.6280v2","created_at":"2026-05-18T02:40:46.937045+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.6280","created_at":"2026-05-18T02:40:46.937045+00:00"},{"alias_kind":"pith_short_12","alias_value":"42EHHOPIEX6F","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_16","alias_value":"42EHHOPIEX6FRKXO","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_8","alias_value":"42EHHOPI","created_at":"2026-05-18T12:28:11.866339+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/42EHHOPIEX6FRKXOFDVRCLBOQG","json":"https://pith.science/pith/42EHHOPIEX6FRKXOFDVRCLBOQG.json","graph_json":"https://pith.science/api/pith-number/42EHHOPIEX6FRKXOFDVRCLBOQG/graph.json","events_json":"https://pith.science/api/pith-number/42EHHOPIEX6FRKXOFDVRCLBOQG/events.json","paper":"https://pith.science/paper/42EHHOPI"},"agent_actions":{"view_html":"https://pith.science/pith/42EHHOPIEX6FRKXOFDVRCLBOQG","download_json":"https://pith.science/pith/42EHHOPIEX6FRKXOFDVRCLBOQG.json","view_paper":"https://pith.science/paper/42EHHOPI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.6280&json=true","fetch_graph":"https://pith.science/api/pith-number/42EHHOPIEX6FRKXOFDVRCLBOQG/graph.json","fetch_events":"https://pith.science/api/pith-number/42EHHOPIEX6FRKXOFDVRCLBOQG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/42EHHOPIEX6FRKXOFDVRCLBOQG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/42EHHOPIEX6FRKXOFDVRCLBOQG/action/storage_attestation","attest_author":"https://pith.science/pith/42EHHOPIEX6FRKXOFDVRCLBOQG/action/author_attestation","sign_citation":"https://pith.science/pith/42EHHOPIEX6FRKXOFDVRCLBOQG/action/citation_signature","submit_replication":"https://pith.science/pith/42EHHOPIEX6FRKXOFDVRCLBOQG/action/replication_record"}},"created_at":"2026-05-18T02:40:46.937045+00:00","updated_at":"2026-05-18T02:40:46.937045+00:00"}