{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:BATNGHKKKJQTS5KSHYSM4E5YGN","short_pith_number":"pith:BATNGHKK","schema_version":"1.0","canonical_sha256":"0826d31d4a52613975523e24ce13b833651146817622010e446e3e45e2341350","source":{"kind":"arxiv","id":"1507.04426","version":1},"attestation_state":"computed","paper":{"title":"Convolution sums of some functions on divisors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Heekyoung Hahn","submitted_at":"2015-07-16T00:57:06Z","abstract_excerpt":"One of the main goals in this paper is to establish convolution sums of functions for the divisor sums $\\widetilde{\\sigma}_s(n)=\\sum_{d|n}(-1)^{d-1}d^s$ and $\\widehat{\\sigma}_s(n)=\\sum_{d|n}(-1)^{\\frac{n}{d}-1}d^s$, for certain $s$, which were first defined by Glaisher. We first introduce three functions $\\mathcal{P}(q)$, $\\mathcal{E}(q)$, and $\\mathcal{Q}(q)$ related to $\\widetilde{\\sigma}(n)$, $\\widehat{\\sigma}(n)$, and $\\widetilde{\\sigma}_3(n)$, respectively, and then we evaluate them in terms of two parameters $x$ and $z$ in Ramanujan's theory of elliptic functions. Using these formulas, w"},"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":"1507.04426","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-16T00:57:06Z","cross_cats_sorted":[],"title_canon_sha256":"84ca14682b90853bd29a67b8cf6c0100821510e707ee4e42e2f5ab1641df0595","abstract_canon_sha256":"87026c4a8438dae73d3baba515ae6278a790d2bb3a87d986df2ecd715725d4e9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:47.420304Z","signature_b64":"D5U9U5SmNJpN+gAbnfE39g9dtSp7l0j4GQspP+LxHkzuFU1KUhmEJM8Crf32i2wrh3L85dz/nBDPiZGmGbRPAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0826d31d4a52613975523e24ce13b833651146817622010e446e3e45e2341350","last_reissued_at":"2026-05-18T01:36:47.419828Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:47.419828Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Convolution sums of some functions on divisors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Heekyoung Hahn","submitted_at":"2015-07-16T00:57:06Z","abstract_excerpt":"One of the main goals in this paper is to establish convolution sums of functions for the divisor sums $\\widetilde{\\sigma}_s(n)=\\sum_{d|n}(-1)^{d-1}d^s$ and $\\widehat{\\sigma}_s(n)=\\sum_{d|n}(-1)^{\\frac{n}{d}-1}d^s$, for certain $s$, which were first defined by Glaisher. We first introduce three functions $\\mathcal{P}(q)$, $\\mathcal{E}(q)$, and $\\mathcal{Q}(q)$ related to $\\widetilde{\\sigma}(n)$, $\\widehat{\\sigma}(n)$, and $\\widetilde{\\sigma}_3(n)$, respectively, and then we evaluate them in terms of two parameters $x$ and $z$ in Ramanujan's theory of elliptic functions. Using these formulas, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04426","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":"1507.04426","created_at":"2026-05-18T01:36:47.419903+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.04426v1","created_at":"2026-05-18T01:36:47.419903+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.04426","created_at":"2026-05-18T01:36:47.419903+00:00"},{"alias_kind":"pith_short_12","alias_value":"BATNGHKKKJQT","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_16","alias_value":"BATNGHKKKJQTS5KS","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_8","alias_value":"BATNGHKK","created_at":"2026-05-18T12:29:14.074870+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/BATNGHKKKJQTS5KSHYSM4E5YGN","json":"https://pith.science/pith/BATNGHKKKJQTS5KSHYSM4E5YGN.json","graph_json":"https://pith.science/api/pith-number/BATNGHKKKJQTS5KSHYSM4E5YGN/graph.json","events_json":"https://pith.science/api/pith-number/BATNGHKKKJQTS5KSHYSM4E5YGN/events.json","paper":"https://pith.science/paper/BATNGHKK"},"agent_actions":{"view_html":"https://pith.science/pith/BATNGHKKKJQTS5KSHYSM4E5YGN","download_json":"https://pith.science/pith/BATNGHKKKJQTS5KSHYSM4E5YGN.json","view_paper":"https://pith.science/paper/BATNGHKK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.04426&json=true","fetch_graph":"https://pith.science/api/pith-number/BATNGHKKKJQTS5KSHYSM4E5YGN/graph.json","fetch_events":"https://pith.science/api/pith-number/BATNGHKKKJQTS5KSHYSM4E5YGN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BATNGHKKKJQTS5KSHYSM4E5YGN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BATNGHKKKJQTS5KSHYSM4E5YGN/action/storage_attestation","attest_author":"https://pith.science/pith/BATNGHKKKJQTS5KSHYSM4E5YGN/action/author_attestation","sign_citation":"https://pith.science/pith/BATNGHKKKJQTS5KSHYSM4E5YGN/action/citation_signature","submit_replication":"https://pith.science/pith/BATNGHKKKJQTS5KSHYSM4E5YGN/action/replication_record"}},"created_at":"2026-05-18T01:36:47.419903+00:00","updated_at":"2026-05-18T01:36:47.419903+00:00"}