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In this paper we study these two zeta functions and related arithmetical functions. We show that $$\\sum^\\infty_{n=1\\atop n\\ \\text{is squarefree}}\\frac{(-e^{2\\pi i/m})^{\\omega(n)}}n=0\\quad\\text{if}\\ \\ m>4,$"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1204.6689","kind":"arxiv","version":13},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-04-30T16:26:38Z","cross_cats_sorted":[],"title_canon_sha256":"d874505eddd2cf3d034bd680361b6d5dc1704869e400448f742908563c6c425a","abstract_canon_sha256":"c127df840777417764e738a67e25a2bab7ecf6d69234d7757f01b59fe0c448d1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:49.757630Z","signature_b64":"lfgHF5ICQ0rhgzwDshdS/nr1unxYG+kSLJ0W0o1jpSIzELidjBCQ1Qehue3n16twb+ZoreWzCtjwDkKGkaxYDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b496463668c651bbcbfafecf0a70700f99b69efc4256bb7c5121ff96d3b3992d","last_reissued_at":"2026-05-18T01:01:49.756966Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:49.756966Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a pair of zeta functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2012-04-30T16:26:38Z","abstract_excerpt":"Let $m$ be a positive integer, and define $$\\zeta_m(s)=\\sum_{n=1}^\\infty\\frac{(-e^{2\\pi i/m})^{\\omega(n)}}{n^s}\\ \\ \\ \\ \\text{and} \\ \\ \\ \\ \\zeta^*_m(s)=\\sum_{n=1}^\\infty\\frac{(-e^{2\\pi i/m})^{\\Omega(n)}}{n^s},$$ for $\\Re(s)>1$, where $\\omega(n)$ denotes the number of distinct prime factors of $n$, and $\\Omega(n)$ represents the total number of prime factors of $n$ (counted with multiplicity). In this paper we study these two zeta functions and related arithmetical functions. We show that $$\\sum^\\infty_{n=1\\atop n\\ \\text{is squarefree}}\\frac{(-e^{2\\pi i/m})^{\\omega(n)}}n=0\\quad\\text{if}\\ \\ m>4,$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.6689","kind":"arxiv","version":13},"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":"1204.6689","created_at":"2026-05-18T01:01:49.757074+00:00"},{"alias_kind":"arxiv_version","alias_value":"1204.6689v13","created_at":"2026-05-18T01:01:49.757074+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.6689","created_at":"2026-05-18T01:01:49.757074+00:00"},{"alias_kind":"pith_short_12","alias_value":"WSLEMNTIYZI3","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_16","alias_value":"WSLEMNTIYZI3XS72","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_8","alias_value":"WSLEMNTI","created_at":"2026-05-18T12:27:27.928770+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/WSLEMNTIYZI3XS7273HQU4DQB6","json":"https://pith.science/pith/WSLEMNTIYZI3XS7273HQU4DQB6.json","graph_json":"https://pith.science/api/pith-number/WSLEMNTIYZI3XS7273HQU4DQB6/graph.json","events_json":"https://pith.science/api/pith-number/WSLEMNTIYZI3XS7273HQU4DQB6/events.json","paper":"https://pith.science/paper/WSLEMNTI"},"agent_actions":{"view_html":"https://pith.science/pith/WSLEMNTIYZI3XS7273HQU4DQB6","download_json":"https://pith.science/pith/WSLEMNTIYZI3XS7273HQU4DQB6.json","view_paper":"https://pith.science/paper/WSLEMNTI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1204.6689&json=true","fetch_graph":"https://pith.science/api/pith-number/WSLEMNTIYZI3XS7273HQU4DQB6/graph.json","fetch_events":"https://pith.science/api/pith-number/WSLEMNTIYZI3XS7273HQU4DQB6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WSLEMNTIYZI3XS7273HQU4DQB6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WSLEMNTIYZI3XS7273HQU4DQB6/action/storage_attestation","attest_author":"https://pith.science/pith/WSLEMNTIYZI3XS7273HQU4DQB6/action/author_attestation","sign_citation":"https://pith.science/pith/WSLEMNTIYZI3XS7273HQU4DQB6/action/citation_signature","submit_replication":"https://pith.science/pith/WSLEMNTIYZI3XS7273HQU4DQB6/action/replication_record"}},"created_at":"2026-05-18T01:01:49.757074+00:00","updated_at":"2026-05-18T01:01:49.757074+00:00"}