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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","authors_text":"Zhi-Hong Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-24T13:00:25Z","title":"Ramanujan's theta functions and sums of triangular numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.06378","kind":"arxiv","version":8},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:e94c0fad4b51f7c9dea3fd5eea5b8ee9361195c98b88d65849c1ff6a441e5d8a","target":"record","created_at":"2026-05-18T00:28:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3b9eba512ed91f3ca6626516803af2b2fa7bc6e24eec4d3e3e134faad5f64ec3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-24T13:00:25Z","title_canon_sha256":"6a04cd6fa27582e384478d64ac2323bed758aa387c41be172aab2a98fe468769"},"schema_version":"1.0","source":{"id":"1601.06378","kind":"arxiv","version":8}},"canonical_sha256":"6883e88808bc36e86fcdbacff9a66eb4a6df77ae364cb9c774596ed347a2e44a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6883e88808bc36e86fcdbacff9a66eb4a6df77ae364cb9c774596ed347a2e44a","first_computed_at":"2026-05-18T00:28:41.427344Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:28:41.427344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1RnZKDNEagXbZAx45bhPKbAbe1020vWORi4zxcvuVgW1ybPDiCjix2JqBO4vUKMqEzsmYEJPSOfqdSnrHo8iCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:28:41.428092Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.06378","source_kind":"arxiv","source_version":8}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e94c0fad4b51f7c9dea3fd5eea5b8ee9361195c98b88d65849c1ff6a441e5d8a","sha256:bedabfa70e5bbddf0afb8cb638f476b639383a4090d164f2ce6dbf43e7e52406"],"state_sha256":"7669a777a4fa0fbbbc40704a06c3c7de5b9dbaaed57c3d88c8912cfe39770806"}