{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:LQRFDOXEVZ67CYHB72FNGL4FYO","short_pith_number":"pith:LQRFDOXE","schema_version":"1.0","canonical_sha256":"5c2251bae4ae7df160e1fe8ad32f85c39e27e8ea8f847846716740ef3e5efbef","source":{"kind":"arxiv","id":"1609.07610","version":1},"attestation_state":"computed","paper":{"title":"On the Sum of Divisors of Mixed Powers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jinjiang Li, Min Zhang","submitted_at":"2016-09-24T12:50:48Z","abstract_excerpt":"Let $d(n)$ denote the Dirichlet divisor function. Define \\begin{equation*}\n  \\mathcal{S}_{k}(x)=\\sum_{\\substack{1\\leqslant n_1,n_2,n_3 \\leqslant x^{1/2} \\\\ 1\\leqslant n_4\\leqslant x^{1/k} }} d(n_1^2+n_2^2+n_3^2+n_4^k),\n  \\qquad 3\\leqslant k\\in \\mathbb{N}. \\end{equation*} In this paper, we establish an asymptotic formula of $\\mathcal{S}_k(x)$ and prove that \\begin{equation*}\n  \\mathcal{S}_k(x)=C_1(k)x^{3/2+1/k}\\log x+C_2(k)x^{3/2+1/k}+O(x^{3/2+1/k-\\delta_k+\\varepsilon}), \\end{equation*} where $C_1(k),\\,C_2(k)$ are two constants depending only on $k,$ with $\\delta_3=\\frac{19}{60},\\,\\delta_4=\\fra"},"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":"1609.07610","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-09-24T12:50:48Z","cross_cats_sorted":[],"title_canon_sha256":"eca877c8c26412ce87c2162779b7ee008a0e3f23fa02b38125fbc0bd5f3ebf99","abstract_canon_sha256":"6f204306c9f2de510e4a6434083f44c94ca957d425d690c31240c0da56b71f1d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:55.916337Z","signature_b64":"iPXMYWFXRq/HFf/vG7KacNDiN6TEuUHnJQRDiZRMfOZ6EuXQ434SGenGW6IQBf2c8sjAKVImoX70GaF0pJbGDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c2251bae4ae7df160e1fe8ad32f85c39e27e8ea8f847846716740ef3e5efbef","last_reissued_at":"2026-05-18T01:03:55.915683Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:55.915683Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Sum of Divisors of Mixed Powers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jinjiang Li, Min Zhang","submitted_at":"2016-09-24T12:50:48Z","abstract_excerpt":"Let $d(n)$ denote the Dirichlet divisor function. Define \\begin{equation*}\n  \\mathcal{S}_{k}(x)=\\sum_{\\substack{1\\leqslant n_1,n_2,n_3 \\leqslant x^{1/2} \\\\ 1\\leqslant n_4\\leqslant x^{1/k} }} d(n_1^2+n_2^2+n_3^2+n_4^k),\n  \\qquad 3\\leqslant k\\in \\mathbb{N}. \\end{equation*} In this paper, we establish an asymptotic formula of $\\mathcal{S}_k(x)$ and prove that \\begin{equation*}\n  \\mathcal{S}_k(x)=C_1(k)x^{3/2+1/k}\\log x+C_2(k)x^{3/2+1/k}+O(x^{3/2+1/k-\\delta_k+\\varepsilon}), \\end{equation*} where $C_1(k),\\,C_2(k)$ are two constants depending only on $k,$ with $\\delta_3=\\frac{19}{60},\\,\\delta_4=\\fra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.07610","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":"1609.07610","created_at":"2026-05-18T01:03:55.915781+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.07610v1","created_at":"2026-05-18T01:03:55.915781+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.07610","created_at":"2026-05-18T01:03:55.915781+00:00"},{"alias_kind":"pith_short_12","alias_value":"LQRFDOXEVZ67","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_16","alias_value":"LQRFDOXEVZ67CYHB","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_8","alias_value":"LQRFDOXE","created_at":"2026-05-18T12:30:29.479603+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/LQRFDOXEVZ67CYHB72FNGL4FYO","json":"https://pith.science/pith/LQRFDOXEVZ67CYHB72FNGL4FYO.json","graph_json":"https://pith.science/api/pith-number/LQRFDOXEVZ67CYHB72FNGL4FYO/graph.json","events_json":"https://pith.science/api/pith-number/LQRFDOXEVZ67CYHB72FNGL4FYO/events.json","paper":"https://pith.science/paper/LQRFDOXE"},"agent_actions":{"view_html":"https://pith.science/pith/LQRFDOXEVZ67CYHB72FNGL4FYO","download_json":"https://pith.science/pith/LQRFDOXEVZ67CYHB72FNGL4FYO.json","view_paper":"https://pith.science/paper/LQRFDOXE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.07610&json=true","fetch_graph":"https://pith.science/api/pith-number/LQRFDOXEVZ67CYHB72FNGL4FYO/graph.json","fetch_events":"https://pith.science/api/pith-number/LQRFDOXEVZ67CYHB72FNGL4FYO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LQRFDOXEVZ67CYHB72FNGL4FYO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LQRFDOXEVZ67CYHB72FNGL4FYO/action/storage_attestation","attest_author":"https://pith.science/pith/LQRFDOXEVZ67CYHB72FNGL4FYO/action/author_attestation","sign_citation":"https://pith.science/pith/LQRFDOXEVZ67CYHB72FNGL4FYO/action/citation_signature","submit_replication":"https://pith.science/pith/LQRFDOXEVZ67CYHB72FNGL4FYO/action/replication_record"}},"created_at":"2026-05-18T01:03:55.915781+00:00","updated_at":"2026-05-18T01:03:55.915781+00:00"}