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For $a_1,a_2,\\ldots,a_k,n\\in\\Bbb N$ let $N(a_1,a_2,\\ldots,a_k;n)$ be the number of representations of $n$ by $a_1x_1^2+a_2x_2^2+\\cdots+a_kx_k^2$, and let $t(a_1,a_2,\\ldots,a_k;n)$ be the number of representations of $n$ by $a_1\\frac{x_1(x_1-1)}2+a_2\\frac{x_2(x_2-1)}2+\\cdots+a_k\\frac{x_k(x_k-1)}2$ $(x_1,\\ldots,x_k\\in\\Bbb Z$). In this paper, by using Ramanujan's theta functions $\\varphi(q)$ and $\\psi(q)$ we reveal many relations between $t(a_1,a_2,\\ldots,a_k;n)$ and $N(a_1,a_2,\\ldots,a_k;8n+a_1+\\cdot"},"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":"1601.06378","kind":"arxiv","version":8},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-24T13:00:25Z","cross_cats_sorted":[],"title_canon_sha256":"6a04cd6fa27582e384478d64ac2323bed758aa387c41be172aab2a98fe468769","abstract_canon_sha256":"3b9eba512ed91f3ca6626516803af2b2fa7bc6e24eec4d3e3e134faad5f64ec3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:28:41.428092Z","signature_b64":"1RnZKDNEagXbZAx45bhPKbAbe1020vWORi4zxcvuVgW1ybPDiCjix2JqBO4vUKMqEzsmYEJPSOfqdSnrHo8iCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6883e88808bc36e86fcdbacff9a66eb4a6df77ae364cb9c774596ed347a2e44a","last_reissued_at":"2026-05-18T00:28:41.427344Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:28:41.427344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ramanujan's theta functions and sums of triangular numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2016-01-24T13:00:25Z","abstract_excerpt":"Let $\\Bbb Z$ and $\\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a_1,a_2,\\ldots,a_k,n\\in\\Bbb N$ let $N(a_1,a_2,\\ldots,a_k;n)$ be the number of representations of $n$ by $a_1x_1^2+a_2x_2^2+\\cdots+a_kx_k^2$, and let $t(a_1,a_2,\\ldots,a_k;n)$ be the number of representations of $n$ by $a_1\\frac{x_1(x_1-1)}2+a_2\\frac{x_2(x_2-1)}2+\\cdots+a_k\\frac{x_k(x_k-1)}2$ $(x_1,\\ldots,x_k\\in\\Bbb Z$). In this paper, by using Ramanujan's theta functions $\\varphi(q)$ and $\\psi(q)$ we reveal many relations between $t(a_1,a_2,\\ldots,a_k;n)$ and $N(a_1,a_2,\\ldots,a_k;8n+a_1+\\cdot"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.06378","kind":"arxiv","version":8},"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":"1601.06378","created_at":"2026-05-18T00:28:41.427470+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.06378v8","created_at":"2026-05-18T00:28:41.427470+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.06378","created_at":"2026-05-18T00:28:41.427470+00:00"},{"alias_kind":"pith_short_12","alias_value":"NCB6RCAIXQ3O","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_16","alias_value":"NCB6RCAIXQ3OQ36N","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_8","alias_value":"NCB6RCAI","created_at":"2026-05-18T12:30:32.724797+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/NCB6RCAIXQ3OQ36NXLH7TJTOWS","json":"https://pith.science/pith/NCB6RCAIXQ3OQ36NXLH7TJTOWS.json","graph_json":"https://pith.science/api/pith-number/NCB6RCAIXQ3OQ36NXLH7TJTOWS/graph.json","events_json":"https://pith.science/api/pith-number/NCB6RCAIXQ3OQ36NXLH7TJTOWS/events.json","paper":"https://pith.science/paper/NCB6RCAI"},"agent_actions":{"view_html":"https://pith.science/pith/NCB6RCAIXQ3OQ36NXLH7TJTOWS","download_json":"https://pith.science/pith/NCB6RCAIXQ3OQ36NXLH7TJTOWS.json","view_paper":"https://pith.science/paper/NCB6RCAI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.06378&json=true","fetch_graph":"https://pith.science/api/pith-number/NCB6RCAIXQ3OQ36NXLH7TJTOWS/graph.json","fetch_events":"https://pith.science/api/pith-number/NCB6RCAIXQ3OQ36NXLH7TJTOWS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NCB6RCAIXQ3OQ36NXLH7TJTOWS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NCB6RCAIXQ3OQ36NXLH7TJTOWS/action/storage_attestation","attest_author":"https://pith.science/pith/NCB6RCAIXQ3OQ36NXLH7TJTOWS/action/author_attestation","sign_citation":"https://pith.science/pith/NCB6RCAIXQ3OQ36NXLH7TJTOWS/action/citation_signature","submit_replication":"https://pith.science/pith/NCB6RCAIXQ3OQ36NXLH7TJTOWS/action/replication_record"}},"created_at":"2026-05-18T00:28:41.427470+00:00","updated_at":"2026-05-18T00:28:41.427470+00:00"}