{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:IVKPPS6O5QMTHJAMWPH6DAATUM","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":"8077a3319c0a1e79cbee8c456a12d0209212043871a84ae22595bcc070e94410","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-01-31T13:41:43Z","title_canon_sha256":"477516e790f973e3d880f414bfe223a69c52a80cc7c2c3c540a894dcbefe02e8"},"schema_version":"1.0","source":{"id":"2202.00637","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2202.00637","created_at":"2026-07-05T04:12:53Z"},{"alias_kind":"arxiv_version","alias_value":"2202.00637v1","created_at":"2026-07-05T04:12:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2202.00637","created_at":"2026-07-05T04:12:53Z"},{"alias_kind":"pith_short_12","alias_value":"IVKPPS6O5QMT","created_at":"2026-07-05T04:12:53Z"},{"alias_kind":"pith_short_16","alias_value":"IVKPPS6O5QMTHJAM","created_at":"2026-07-05T04:12:53Z"},{"alias_kind":"pith_short_8","alias_value":"IVKPPS6O","created_at":"2026-07-05T04:12:53Z"}],"graph_snapshots":[{"event_id":"sha256:d1a7bb2f2af6353cfecc2882a973424a51ed5c02877ee77ca29458dd3d4080ef","target":"graph","created_at":"2026-07-05T04:12:53Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2202.00637/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In 1916, Riesz proved that the Riemann hypothesis is equivalent to the bound $\\sum_{n=1}^\\infty \\frac{\\mu(n)}{n^2} \\exp\\left( - \\frac{x}{n^2} \\right) = O_{\\epsilon} \\left( x^{-\\frac{3}{4} + \\epsilon} \\right)$, as $x \\rightarrow\\infty$, for any $\\epsilon >0$. Around the same time, Hardy and Littlewood gave another equivalent criteria for the Riemann hypothesis while correcting an identity of Ramanujan. In the present paper, we establish a one-variable generalization of the identity of Hardy and Littlewood and as an application, we provide Riesz-type criteria for the Riemann hypothesis. In parti","authors_text":"Archit Agarwal, Bibekananda Maji, Meghali Garg","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-01-31T13:41:43Z","title":"Riesz-type criteria for the Riemann hypothesis"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2202.00637","kind":"arxiv","version":1},"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:fb7250523b299e1f8f8d9f868fb8114b801ec41e3f25bd5d8704eeb86d7d2c07","target":"record","created_at":"2026-07-05T04:12:53Z","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":"8077a3319c0a1e79cbee8c456a12d0209212043871a84ae22595bcc070e94410","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-01-31T13:41:43Z","title_canon_sha256":"477516e790f973e3d880f414bfe223a69c52a80cc7c2c3c540a894dcbefe02e8"},"schema_version":"1.0","source":{"id":"2202.00637","kind":"arxiv","version":1}},"canonical_sha256":"4554f7cbceec1933a40cb3cfe18013a320762e70807a2529c19518c2cca7ff36","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4554f7cbceec1933a40cb3cfe18013a320762e70807a2529c19518c2cca7ff36","first_computed_at":"2026-07-05T04:12:53.120439Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:12:53.120439Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Bs/HaS3eDojQYB7LHVNtougprN2nYEeVaMQbRMrkZS6oG5cEkIJyxECdjdSd1F3XRgLy/nGCH67JsYZSBWDaDw==","signature_status":"signed_v1","signed_at":"2026-07-05T04:12:53.120993Z","signed_message":"canonical_sha256_bytes"},"source_id":"2202.00637","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb7250523b299e1f8f8d9f868fb8114b801ec41e3f25bd5d8704eeb86d7d2c07","sha256:d1a7bb2f2af6353cfecc2882a973424a51ed5c02877ee77ca29458dd3d4080ef"],"state_sha256":"f1cd9d40652f0bb43756bcb7bad5430da626693efb1195300fd136f5653bfa29"}