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Ramakrishnan, Brundaban Sahu","submitted_at":"2012-07-21T07:11:53Z","abstract_excerpt":"We evaluate the convolution sums $\\sum_{l,m\\in {\\mathbb N}, {l+15m=n}} \\sigma(l) \\sigma(m)$ and $\\sum_{l,m\\in {\\mathbb N}, {3l+5m=n}} \\sigma(l) \\sigma(m)$ for all $n\\in {\\mathbb N}$ using the theory of quasimodular forms and use these convolution sums to determine the number of representations of a positive integer $n$ by the form $$ x_1^2 + x_1x_2 + x_2^2 + x_3^2 + x_3x_4 + x_4^2 + 5 (x_5^2 + x_5x_6 + x_6^2 + x_7^2 + x_7x_8 + x_8^2). $$ We also determine the number of representations of positive integers by the quadratic form $$ x_1^2 + x_2^2+x_3^2+x_4^2 + 6 (x_5^2+x_6^2+x_7^2+x_8^2), $$ by u"},"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":"1207.5107","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-07-21T07:11:53Z","cross_cats_sorted":[],"title_canon_sha256":"30640f459a676dfd6c18c9876fe12d8e2d69a9e1b95e90401d25178ce9392f71","abstract_canon_sha256":"a5630f1955b26ac90cceefcffad80e6973ae58f38f1861efeff644f7bcd1176c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:42:44.089685Z","signature_b64":"tHy4p58RSBgC6bdl5Mnt4es2ar6LB5GulQiZLM76x+g85yuBB5RplEABASv75aZT9OYAopeO1i2poFgZNvksAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"51b968f31873d74922489bf5bbbf27eb7c5a1ba4ea844955083f053d7021e619","last_reissued_at":"2026-05-18T03:42:44.089156Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:42:44.089156Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Evaluation of the convolution sums $\\sum_{l+15m=n} \\sigma(l) \\sigma(m)$ and $\\sum_{3l+5m=n} \\sigma(l) \\sigma(m)$ and some applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"B. 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