{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:F5OKTT56OM4GJU2WL44OMCHDZI","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":"c673cb8ec10b6339184bd39d32704a58dbf424691f221ed6be82f30638ea95d6","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-23T18:59:58Z","title_canon_sha256":"ff39edca1238c538fe5ef39ee8525119c9ed2eb3d91476ec69537e793c520d0f"},"schema_version":"1.0","source":{"id":"2606.25094","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.25094","created_at":"2026-06-25T00:18:17Z"},{"alias_kind":"arxiv_version","alias_value":"2606.25094v1","created_at":"2026-06-25T00:18:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.25094","created_at":"2026-06-25T00:18:17Z"},{"alias_kind":"pith_short_12","alias_value":"F5OKTT56OM4G","created_at":"2026-06-25T00:18:17Z"},{"alias_kind":"pith_short_16","alias_value":"F5OKTT56OM4GJU2W","created_at":"2026-06-25T00:18:17Z"},{"alias_kind":"pith_short_8","alias_value":"F5OKTT56","created_at":"2026-06-25T00:18:17Z"}],"graph_snapshots":[{"event_id":"sha256:ae56b787508bd6c61a35e323a790af28d1d8a5e15f382a3938ee5c1fc185f0f9","target":"graph","created_at":"2026-06-25T00:18:17Z","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/2606.25094/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $\\chi$ be a primitive Dirichlet character modulo $q>1$. Assuming the Generalised Riemann Hypothesis for $L(s,\\chi)$ and that the non-trivial zeros $\\rho=\\tfrac12+i\\gamma$ of $L(s,\\chi)$ are simple, we prove lower bounds for the discrete moments $\\sum_{0<\\gamma\\le T}|L'(\\rho,\\chi)|^{-2}$ and $\\sum_{0<\\gamma\\le T}|L(2\\rho,\\chi^2)/L'(\\rho,\\chi)|^2$, uniformly in the conductor. The bounds capture the proportion $\\beta/(1+\\beta)$ of the conjectured asymptotics, where $\\beta=\\log T/\\log qT$: this is one half whenever $\\log q=o(\\log T)$, recovering for fixed $q$ the Dirichlet analogues of theorem","authors_text":"Andrew Pearce-Crump","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-23T18:59:58Z","title":"Negative discrete second moments of Dirichlet $L$-functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.25094","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:c26d123b12ce4c0debe3b18e90acad5c54ad2f882c87333b6d46c4608f1123f5","target":"record","created_at":"2026-06-25T00:18:17Z","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":"c673cb8ec10b6339184bd39d32704a58dbf424691f221ed6be82f30638ea95d6","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-23T18:59:58Z","title_canon_sha256":"ff39edca1238c538fe5ef39ee8525119c9ed2eb3d91476ec69537e793c520d0f"},"schema_version":"1.0","source":{"id":"2606.25094","kind":"arxiv","version":1}},"canonical_sha256":"2f5ca9cfbe733864d3565f38e608e3ca250bdc644cfb6822f407e90269dfbdf9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2f5ca9cfbe733864d3565f38e608e3ca250bdc644cfb6822f407e90269dfbdf9","first_computed_at":"2026-06-25T00:18:17.779430Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-25T00:18:17.779430Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IaPYcWaHBltFJyZu2poHhk6EAujZ+Goy2XMBDjwe4l+v9OngIHQ/HorOINwXZ7CjBqW47NIJv5rlNELIHbYBCg==","signature_status":"signed_v1","signed_at":"2026-06-25T00:18:17.779805Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.25094","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c26d123b12ce4c0debe3b18e90acad5c54ad2f882c87333b6d46c4608f1123f5","sha256:ae56b787508bd6c61a35e323a790af28d1d8a5e15f382a3938ee5c1fc185f0f9"],"state_sha256":"74ef1101da30bb711b0d39af42ecac7a3c33f895d842caaedcf9500660572efe"}