{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:UKK36SKGM2JWVFY5EY4QP7NH2F","short_pith_number":"pith:UKK36SKG","schema_version":"1.0","canonical_sha256":"a295bf494666936a971d263907fda7d154f8a6cca4ca8071c998da054a0b6c3c","source":{"kind":"arxiv","id":"2304.00943","version":2},"attestation_state":"computed","paper":{"title":"Almost sure upper bound for random multiplicative functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Rachid Caich","submitted_at":"2023-04-03T13:00:31Z","abstract_excerpt":"Let $\\varepsilon >0$. Let $f$ be a Steinhaus or Rademacher random multiplicative function. We prove that we have almost surely, as $x \\to +\\infty$,\n  $$\n  \\sum_{n \\leqslant x} f(n) \\ll \\sqrt{x} (\\log_2 x)^{\\frac{3}{4}+ \\varepsilon}.\n  $$"},"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":"2304.00943","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2023-04-03T13:00:31Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"46489ceccfa222a2e189ee7d86c230ced05ce2110c404fe1d529d9e83847e2e1","abstract_canon_sha256":"998c77ac97b32952cf6a2a015635a2cc8b09696095c537ce2ef915976d436e4e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:56:12.227436Z","signature_b64":"bccuC1rgQt84lX+nIfysOI/67vMNJoaWoeFkOT2n7ol1Rr8q9Nqyy+Qxb98CXIb//o+Ub0iT199QLJ2htfs8CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a295bf494666936a971d263907fda7d154f8a6cca4ca8071c998da054a0b6c3c","last_reissued_at":"2026-07-05T08:56:12.226698Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:56:12.226698Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Almost sure upper bound for random multiplicative functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Rachid Caich","submitted_at":"2023-04-03T13:00:31Z","abstract_excerpt":"Let $\\varepsilon >0$. Let $f$ be a Steinhaus or Rademacher random multiplicative function. We prove that we have almost surely, as $x \\to +\\infty$,\n  $$\n  \\sum_{n \\leqslant x} f(n) \\ll \\sqrt{x} (\\log_2 x)^{\\frac{3}{4}+ \\varepsilon}.\n  $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2304.00943","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2304.00943/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2304.00943","created_at":"2026-07-05T08:56:12.226782+00:00"},{"alias_kind":"arxiv_version","alias_value":"2304.00943v2","created_at":"2026-07-05T08:56:12.226782+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2304.00943","created_at":"2026-07-05T08:56:12.226782+00:00"},{"alias_kind":"pith_short_12","alias_value":"UKK36SKGM2JW","created_at":"2026-07-05T08:56:12.226782+00:00"},{"alias_kind":"pith_short_16","alias_value":"UKK36SKGM2JWVFY5","created_at":"2026-07-05T08:56:12.226782+00:00"},{"alias_kind":"pith_short_8","alias_value":"UKK36SKG","created_at":"2026-07-05T08:56:12.226782+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2608.09906","citing_title":"On the $\\beta=2$ Partition function for Dirichlet $L$-functions in the $q$-aspect","ref_index":4,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UKK36SKGM2JWVFY5EY4QP7NH2F","json":"https://pith.science/pith/UKK36SKGM2JWVFY5EY4QP7NH2F.json","graph_json":"https://pith.science/api/pith-number/UKK36SKGM2JWVFY5EY4QP7NH2F/graph.json","events_json":"https://pith.science/api/pith-number/UKK36SKGM2JWVFY5EY4QP7NH2F/events.json","paper":"https://pith.science/paper/UKK36SKG"},"agent_actions":{"view_html":"https://pith.science/pith/UKK36SKGM2JWVFY5EY4QP7NH2F","download_json":"https://pith.science/pith/UKK36SKGM2JWVFY5EY4QP7NH2F.json","view_paper":"https://pith.science/paper/UKK36SKG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2304.00943&json=true","fetch_graph":"https://pith.science/api/pith-number/UKK36SKGM2JWVFY5EY4QP7NH2F/graph.json","fetch_events":"https://pith.science/api/pith-number/UKK36SKGM2JWVFY5EY4QP7NH2F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UKK36SKGM2JWVFY5EY4QP7NH2F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UKK36SKGM2JWVFY5EY4QP7NH2F/action/storage_attestation","attest_author":"https://pith.science/pith/UKK36SKGM2JWVFY5EY4QP7NH2F/action/author_attestation","sign_citation":"https://pith.science/pith/UKK36SKGM2JWVFY5EY4QP7NH2F/action/citation_signature","submit_replication":"https://pith.science/pith/UKK36SKGM2JWVFY5EY4QP7NH2F/action/replication_record"}},"created_at":"2026-07-05T08:56:12.226782+00:00","updated_at":"2026-07-05T08:56:12.226782+00:00"}