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We further obtain that the Ces\\`{a}ro mean of the self-correlations and some moving average of the self-correlations of such multiplicative functions converge to zero. Our proof gives, for any $N \\geq 2$, $$\\frac1{N}\\sum_{m=1}^{N}\\Big|\\frac1{N}\\sum_{n=1}^{N} \\bnu(n) \\bnu(n+m)\\Big| \\leq \\frac{C}{\\log(N)^{\\epsilon}},$$ and $$\\frac1{N^2}\\sum_{n,p=1}^{N}\\Big|\\frac1{N"},"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":"1606.05630","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-06-17T19:25:50Z","cross_cats_sorted":["math.CO","math.CV","math.NT","math.PR"],"title_canon_sha256":"52d34decf6070097e5beaea54a362f3c0e468553a54d2256415985ab6e04e937","abstract_canon_sha256":"e6ab17308b67cebf16369e741caae849a64116c1da45dfb1016bfc66ea3b2fda"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:00:12.263798Z","signature_b64":"wkXIhiMPbCFT1JH9DiLqS2HESNHX+D/1vijiWms3bR82om8GYCeo32vz/OqfyuPTQrWh41tPjEj5S4DDBqvjCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e89ecd1a6bc8e15c7a969a30fd09a56394f7583847f4a2ee6440e02994bfadb3","last_reissued_at":"2026-05-18T01:00:12.263096Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:00:12.263096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A cubic nonconventional ergodic average with multiplicative or Mangoldt weights","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.CV","math.NT","math.PR"],"primary_cat":"math.DS","authors_text":"el Houcein el Abdalaoui, Xiangdong Ye","submitted_at":"2016-06-17T19:25:50Z","abstract_excerpt":"We show that the cubic nonconventional ergodic averages of any order with a bounded multiplicative function weight converge almost surely to zero provided that the multiplicative function satisfies a strong Daboussi-Delange condition. 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