{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:5OU4KZZHIGAODU3PKTLULLPQF4","short_pith_number":"pith:5OU4KZZH","schema_version":"1.0","canonical_sha256":"eba9c567274180e1d36f54d745adf02f11a8e743c67ff8ed1fad590aba71ca00","source":{"kind":"arxiv","id":"1703.08785","version":2},"attestation_state":"computed","paper":{"title":"The $r$th moment of the divisor function: an elementary approach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Florian Luca, L\\'aszl\\'o T\\'oth","submitted_at":"2017-03-26T08:39:01Z","abstract_excerpt":"Let $\\tau(n)$ be the number of divisors of $n$. We give an elementary proof of the fact that $$ \\sum_{n\\le x} \\tau(n)^r =xC_{r} (\\log x)^{2^r-1}+O(x(\\log x)^{2^r-2}), $$ for any integer $r\\ge 2$. Here, $$ C_{r}=\\frac{1}{(2^r-1)!} \\prod_{p\\ge 2}\\left( \\left(1-\\frac{1}{p}\\right)^{2^r} \\left(\\sum_{\\alpha\\ge 0} \\frac{(\\alpha+1)^r}{p^{\\alpha}}\\right)\\right). $$"},"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":"1703.08785","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-03-26T08:39:01Z","cross_cats_sorted":[],"title_canon_sha256":"e419933886c8b46a20f1beec3d2a93ec56af886e31934f4d7d322395739d60c2","abstract_canon_sha256":"9ee85faf7ee747517d142d9f28798a837bdaec5cee678b30d29814a458748bdf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:58.397003Z","signature_b64":"7wY30uNMiAhN/Vqv/bUhno/+uqpWU+vH4684ToMMa7MZa8ofDd4ZMyUaHKHwqEtiI9DH/dcFLwIN9M2PQo5FAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eba9c567274180e1d36f54d745adf02f11a8e743c67ff8ed1fad590aba71ca00","last_reissued_at":"2026-05-18T00:40:58.396395Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:58.396395Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The $r$th moment of the divisor function: an elementary approach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Florian Luca, L\\'aszl\\'o T\\'oth","submitted_at":"2017-03-26T08:39:01Z","abstract_excerpt":"Let $\\tau(n)$ be the number of divisors of $n$. We give an elementary proof of the fact that $$ \\sum_{n\\le x} \\tau(n)^r =xC_{r} (\\log x)^{2^r-1}+O(x(\\log x)^{2^r-2}), $$ for any integer $r\\ge 2$. Here, $$ C_{r}=\\frac{1}{(2^r-1)!} \\prod_{p\\ge 2}\\left( \\left(1-\\frac{1}{p}\\right)^{2^r} \\left(\\sum_{\\alpha\\ge 0} \\frac{(\\alpha+1)^r}{p^{\\alpha}}\\right)\\right). $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.08785","kind":"arxiv","version":2},"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":"1703.08785","created_at":"2026-05-18T00:40:58.396486+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.08785v2","created_at":"2026-05-18T00:40:58.396486+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.08785","created_at":"2026-05-18T00:40:58.396486+00:00"},{"alias_kind":"pith_short_12","alias_value":"5OU4KZZHIGAO","created_at":"2026-05-18T12:31:00.734936+00:00"},{"alias_kind":"pith_short_16","alias_value":"5OU4KZZHIGAODU3P","created_at":"2026-05-18T12:31:00.734936+00:00"},{"alias_kind":"pith_short_8","alias_value":"5OU4KZZH","created_at":"2026-05-18T12:31:00.734936+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/5OU4KZZHIGAODU3PKTLULLPQF4","json":"https://pith.science/pith/5OU4KZZHIGAODU3PKTLULLPQF4.json","graph_json":"https://pith.science/api/pith-number/5OU4KZZHIGAODU3PKTLULLPQF4/graph.json","events_json":"https://pith.science/api/pith-number/5OU4KZZHIGAODU3PKTLULLPQF4/events.json","paper":"https://pith.science/paper/5OU4KZZH"},"agent_actions":{"view_html":"https://pith.science/pith/5OU4KZZHIGAODU3PKTLULLPQF4","download_json":"https://pith.science/pith/5OU4KZZHIGAODU3PKTLULLPQF4.json","view_paper":"https://pith.science/paper/5OU4KZZH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.08785&json=true","fetch_graph":"https://pith.science/api/pith-number/5OU4KZZHIGAODU3PKTLULLPQF4/graph.json","fetch_events":"https://pith.science/api/pith-number/5OU4KZZHIGAODU3PKTLULLPQF4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5OU4KZZHIGAODU3PKTLULLPQF4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5OU4KZZHIGAODU3PKTLULLPQF4/action/storage_attestation","attest_author":"https://pith.science/pith/5OU4KZZHIGAODU3PKTLULLPQF4/action/author_attestation","sign_citation":"https://pith.science/pith/5OU4KZZHIGAODU3PKTLULLPQF4/action/citation_signature","submit_replication":"https://pith.science/pith/5OU4KZZHIGAODU3PKTLULLPQF4/action/replication_record"}},"created_at":"2026-05-18T00:40:58.396486+00:00","updated_at":"2026-05-18T00:40:58.396486+00:00"}