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Define the arithmetic function $f_k(a,b;c;n)$ by $$ \\sum_{n=1}^{\\infty}\\frac{f_k(a,b;c;n)}{n^s}=\\frac{\\zeta (as)\\zeta (bs)}{\\zeta^k(cs)}, \\Re s >1.$$\n  Let $\\Delta_k(a,b;c;x)$ denote the error term of the summatory function of the function $f_k(a,b;c;n).$ IN this paper we shall give two expressions of $\\Delta_k(a,b;c;x)$. As applications, we study the so-called $(l,r)$-integers, the generalized square-full integers, the $e-r$-free integers, the divisor problem over $r$-free integers, the $e"},"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":"1301.4530","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-01-19T04:24:55Z","cross_cats_sorted":[],"title_canon_sha256":"9aa5ccfea2db513d12a91a4e4555740ada305cd69b33144eef8bd7c76f7f908b","abstract_canon_sha256":"9a5377f8cf9b68bf9f2d72edbf7a938371b6823eea201a8b8f3a4622b7385f97"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:35:56.459087Z","signature_b64":"ckyEN419aKZmtd6qlxTrXpXqHisbE44Uhw76BJ3r+WoUyOc0T0zKRId9BoP9pGFw5egTBUbOcr5MkO2ECAXMAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e5fa573c49eb4f0936e8feb3b4637d68a111972548a532a6b3605231533d0272","last_reissued_at":"2026-05-18T03:35:56.458448Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:35:56.458448Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Conditional Results for a Class of Arithmetic Functions: a variant of H. 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As applications, we study the so-called $(l,r)$-integers, the generalized square-full integers, the $e-r$-free integers, the divisor problem over $r$-free integers, the $e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.4530","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":"1301.4530","created_at":"2026-05-18T03:35:56.458560+00:00"},{"alias_kind":"arxiv_version","alias_value":"1301.4530v1","created_at":"2026-05-18T03:35:56.458560+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.4530","created_at":"2026-05-18T03:35:56.458560+00:00"},{"alias_kind":"pith_short_12","alias_value":"4X5FOPCJ5NHQ","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_16","alias_value":"4X5FOPCJ5NHQSNXI","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_8","alias_value":"4X5FOPCJ","created_at":"2026-05-18T12:27:34.582898+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/4X5FOPCJ5NHQSNXI72Z3IY35NC","json":"https://pith.science/pith/4X5FOPCJ5NHQSNXI72Z3IY35NC.json","graph_json":"https://pith.science/api/pith-number/4X5FOPCJ5NHQSNXI72Z3IY35NC/graph.json","events_json":"https://pith.science/api/pith-number/4X5FOPCJ5NHQSNXI72Z3IY35NC/events.json","paper":"https://pith.science/paper/4X5FOPCJ"},"agent_actions":{"view_html":"https://pith.science/pith/4X5FOPCJ5NHQSNXI72Z3IY35NC","download_json":"https://pith.science/pith/4X5FOPCJ5NHQSNXI72Z3IY35NC.json","view_paper":"https://pith.science/paper/4X5FOPCJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1301.4530&json=true","fetch_graph":"https://pith.science/api/pith-number/4X5FOPCJ5NHQSNXI72Z3IY35NC/graph.json","fetch_events":"https://pith.science/api/pith-number/4X5FOPCJ5NHQSNXI72Z3IY35NC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4X5FOPCJ5NHQSNXI72Z3IY35NC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4X5FOPCJ5NHQSNXI72Z3IY35NC/action/storage_attestation","attest_author":"https://pith.science/pith/4X5FOPCJ5NHQSNXI72Z3IY35NC/action/author_attestation","sign_citation":"https://pith.science/pith/4X5FOPCJ5NHQSNXI72Z3IY35NC/action/citation_signature","submit_replication":"https://pith.science/pith/4X5FOPCJ5NHQSNXI72Z3IY35NC/action/replication_record"}},"created_at":"2026-05-18T03:35:56.458560+00:00","updated_at":"2026-05-18T03:35:56.458560+00:00"}