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We consider the correlation $$ \\langle a(n)A(n-1) \\rangle (T) = \\frac{1}{\\zeta(1+\\delta(T))}\\sum_{n\\leq T^{1-c}}\\frac{a(n)A(n-1)}{n^{1+\\delta(T)}} $$ where $0<c<1$ is arbitrary and $0<\\delta(T)=O\\left(T^{c-1}\\right)$ is suitably chosen. Let $\\mu(.)$ and $\\lambda(.)$ denote the M\\\"obius function and the Liouville function respectively while $M(.)$ and $L(.)$ denote their corresponding summatory functions. Under the Riemann hypothesis a"},"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":"2409.02106","kind":"arxiv","version":10},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-09-03T17:58:50Z","cross_cats_sorted":[],"title_canon_sha256":"dfe22402efca8115cffaddd2f383eeb10e0c88a1c7972e45f0c8a2a034afe68d","abstract_canon_sha256":"b0a0ac48ea09bbbeb1a0047c12bc3d7a7b3e666cadbab12104b1473bba92e21b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:01.746906Z","signature_b64":"Vmkp7dX8gr5OT3dO0rjPh1UXeqivSR4uobQjfH6oi8FtOGLu0FZeGhY6uOeJ4+oySB5PyQZs3LzJxSVEhfHEBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ff7fa29f5dccd1ef65cf1e2f7e0983bce0a8da45bd98ab7abbb979c4654e570c","last_reissued_at":"2026-05-17T23:39:01.746338Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:01.746338Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Correlations of multiplicative functions with their partial sums","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Gordon Chavez","submitted_at":"2024-09-03T17:58:50Z","abstract_excerpt":"Let $\\zeta(.)$ denote the Riemann zeta function and let $a(.)$ and $A(.)$ respectively denote a multiplicative function and its corresponding summatory function. We consider the correlation $$ \\langle a(n)A(n-1) \\rangle (T) = \\frac{1}{\\zeta(1+\\delta(T))}\\sum_{n\\leq T^{1-c}}\\frac{a(n)A(n-1)}{n^{1+\\delta(T)}} $$ where $0<c<1$ is arbitrary and $0<\\delta(T)=O\\left(T^{c-1}\\right)$ is suitably chosen. Let $\\mu(.)$ and $\\lambda(.)$ denote the M\\\"obius function and the Liouville function respectively while $M(.)$ and $L(.)$ denote their corresponding summatory functions. 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