{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:IEWF2LWSQLUEL24DGD5KJRFA4X","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"24398044ef532f68faea1f6329740c8fdca2c7c44b277b284704489be1bc1bff","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-05-19T13:13:02Z","title_canon_sha256":"a8d84e771e101d526eb4a1a08fe47b65e6154ac9063fe2eaf5c4f58626719ae4"},"schema_version":"1.0","source":{"id":"1305.4361","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.4361","created_at":"2026-05-18T03:20:22Z"},{"alias_kind":"arxiv_version","alias_value":"1305.4361v2","created_at":"2026-05-18T03:20:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.4361","created_at":"2026-05-18T03:20:22Z"},{"alias_kind":"pith_short_12","alias_value":"IEWF2LWSQLUE","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"IEWF2LWSQLUEL24D","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"IEWF2LWS","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:c43e90dcd8edadd29c6e9791803c5cdbc14fae6b7d5847274ef7f08300ba38a2","target":"graph","created_at":"2026-05-18T03:20:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Assuming Sarnak conjecture is true for any singular dynamical process, we prove that the spectral measure of the M\\\"{o}bius function is equivalent to Lebesgue measure. Conversely, under Elliott conjecture, we establish that the M\\\"{o}bius function is orthogonal to any uniquely ergodic dynamical system with singular spectrum. Furthermore, using Mirsky Theorem, we find a new simple proof of Cellarosi-Sinai Theorem on the orthogonality of the square of the M\\\"{o}bius function with respect to any weakly mixing dynamical system. Finally, we establish Sarnak conjecture for a particular random model.","authors_text":"E. H. El Abdalaoui, M. Disertori","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-05-19T13:13:02Z","title":"Spectral properties of the M\\\"{o}bius function and a random M\\\"{o}bius model"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.4361","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:634a10a9af266d491ccb217ddbaa981b5a4bf2b0533af1f01d74994f6f6fac75","target":"record","created_at":"2026-05-18T03:20:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"24398044ef532f68faea1f6329740c8fdca2c7c44b277b284704489be1bc1bff","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-05-19T13:13:02Z","title_canon_sha256":"a8d84e771e101d526eb4a1a08fe47b65e6154ac9063fe2eaf5c4f58626719ae4"},"schema_version":"1.0","source":{"id":"1305.4361","kind":"arxiv","version":2}},"canonical_sha256":"412c5d2ed282e845eb8330faa4c4a0e5f99b163e6d007885232b9e5569baf58b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"412c5d2ed282e845eb8330faa4c4a0e5f99b163e6d007885232b9e5569baf58b","first_computed_at":"2026-05-18T03:20:22.198061Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:20:22.198061Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"S6/OGHW+WGxPEkscAL4T/7aYMTMYBKdLimkWU5wlAqko/QuBCVbXqEiZfBpdixGCV/Fd8kCyrB8GTRek2v5hCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:20:22.198800Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.4361","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:634a10a9af266d491ccb217ddbaa981b5a4bf2b0533af1f01d74994f6f6fac75","sha256:c43e90dcd8edadd29c6e9791803c5cdbc14fae6b7d5847274ef7f08300ba38a2"],"state_sha256":"45a60a6e0a9d54c17f20c0e46059da351b421402367840d63069f227e0a47ffa"}