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Assuming an asymptotic behavior of summatory function, F{f;N,1}\\stackrel{N\\to \\infty}{=}G_1(N) [1+ {\\cal O}(G_2(N))], where G_1(N)=N^{a_1}(log N)^{b_1}, G_2(N)=N^{-a_2}(log N)^{-b_2} and a_1, a_2\\geq 0, -\\infty < b_1, b_2< \\infty, we calculate a renormalization function defined as a ratio, R(f;N,p^m)=F{f;N,p^m}/F{f;N,1}, and find its asymptotics R_{\\infty}(f;p^m) w"},"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":"1108.0957","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-03T22:59:07Z","cross_cats_sorted":[],"title_canon_sha256":"6d9ffcc076e7c389bb4b6cc34ac0e924baf3ea8237f03ae59a2c0c12c7a69923","abstract_canon_sha256":"58560f1a688139c8d9d1bbaf6377c3a63142a607e7872f9dd54deab65091d47c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:16:13.318793Z","signature_b64":"v+h0yU03R24Pw2q+HEV6UwFRK2+33yIMGaCegQ+6+9sHeQ/OL67KP6E7m5XNG+QIPKwwlxqFaejPGIP0bivzCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"39ba4f37c34fcffe65270109c13d8b2707ae23fbaa0a6a6bd3a9139f5ad3dc21","last_reissued_at":"2026-05-18T04:16:13.318121Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:16:13.318121Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Leonid G. Fel","submitted_at":"2011-08-03T22:59:07Z","abstract_excerpt":"We consider a wide class of summatory functions F{f;N,p^m}=\\sum_{k\\leq N}f(p^m k), m\\in \\mathbb Z_+\\cup {0}, associated with the multiplicative arithmetic functions f of a scaled variable k\\in \\mathbb Z_+, where p is a prime number. Assuming an asymptotic behavior of summatory function, F{f;N,1}\\stackrel{N\\to \\infty}{=}G_1(N) [1+ {\\cal O}(G_2(N))], where G_1(N)=N^{a_1}(log N)^{b_1}, G_2(N)=N^{-a_2}(log N)^{-b_2} and a_1, a_2\\geq 0, -\\infty < b_1, b_2< \\infty, we calculate a renormalization function defined as a ratio, R(f;N,p^m)=F{f;N,p^m}/F{f;N,1}, and find its asymptotics R_{\\infty}(f;p^m) w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.0957","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":"1108.0957","created_at":"2026-05-18T04:16:13.318236+00:00"},{"alias_kind":"arxiv_version","alias_value":"1108.0957v1","created_at":"2026-05-18T04:16:13.318236+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.0957","created_at":"2026-05-18T04:16:13.318236+00:00"},{"alias_kind":"pith_short_12","alias_value":"HG5E6N6DJ7H7","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_16","alias_value":"HG5E6N6DJ7H74ZJH","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_8","alias_value":"HG5E6N6D","created_at":"2026-05-18T12:26:30.835961+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/HG5E6N6DJ7H74ZJHAEE4CPMLE4","json":"https://pith.science/pith/HG5E6N6DJ7H74ZJHAEE4CPMLE4.json","graph_json":"https://pith.science/api/pith-number/HG5E6N6DJ7H74ZJHAEE4CPMLE4/graph.json","events_json":"https://pith.science/api/pith-number/HG5E6N6DJ7H74ZJHAEE4CPMLE4/events.json","paper":"https://pith.science/paper/HG5E6N6D"},"agent_actions":{"view_html":"https://pith.science/pith/HG5E6N6DJ7H74ZJHAEE4CPMLE4","download_json":"https://pith.science/pith/HG5E6N6DJ7H74ZJHAEE4CPMLE4.json","view_paper":"https://pith.science/paper/HG5E6N6D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1108.0957&json=true","fetch_graph":"https://pith.science/api/pith-number/HG5E6N6DJ7H74ZJHAEE4CPMLE4/graph.json","fetch_events":"https://pith.science/api/pith-number/HG5E6N6DJ7H74ZJHAEE4CPMLE4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HG5E6N6DJ7H74ZJHAEE4CPMLE4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HG5E6N6DJ7H74ZJHAEE4CPMLE4/action/storage_attestation","attest_author":"https://pith.science/pith/HG5E6N6DJ7H74ZJHAEE4CPMLE4/action/author_attestation","sign_citation":"https://pith.science/pith/HG5E6N6DJ7H74ZJHAEE4CPMLE4/action/citation_signature","submit_replication":"https://pith.science/pith/HG5E6N6DJ7H74ZJHAEE4CPMLE4/action/replication_record"}},"created_at":"2026-05-18T04:16:13.318236+00:00","updated_at":"2026-05-18T04:16:13.318236+00:00"}