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Conditioned on the Riemann hypothesis, we show that the probability this count is greater than $x$ decays at least as quickly as $e^{-Cx\\log x}$, uniformly in $T$. We also prove a similar results for the logarithmic derivative of the zeta function, and likewise analogous results for the eigenvalues of a random unitary matrix.\n  We use results of this sort to show on the Riemann hypothesis that the av"},"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":"1502.05658","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-02-19T18:05:51Z","cross_cats_sorted":[],"title_canon_sha256":"bdee2aac9e09c912e20487bd16c70b9d09c5cdda98dd6f9b9a2693f0dc2b781e","abstract_canon_sha256":"8de010aee64add7293e61f93b5ad008417ce52956761abe80be0afdb33499e84"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:18.824153Z","signature_b64":"1bLwQ82DJ8+aoHKwLw7dERNcgiuOp1hj2RX14KYNcwiyZUhCEmLQ4ny9d2buYgZK8qoBf5Xhzx6aG4Y7WnlLAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4be49825723fc00af186ae8fc05b84abb102b76a76b9b2c6c1da0743c016b549","last_reissued_at":"2026-05-18T00:35:18.823766Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:18.823766Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tail bounds for counts of zeros and eigenvalues, and an application to ratios","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Brad Rodgers","submitted_at":"2015-02-19T18:05:51Z","abstract_excerpt":"Let $t$ be random and uniformly distributed in the interval $[T,2T]$, and consider the quantity $N(t+1/\\log T) - N(t)$, a count of zeros of the Riemann zeta function in a box of height $1/\\log T$. Conditioned on the Riemann hypothesis, we show that the probability this count is greater than $x$ decays at least as quickly as $e^{-Cx\\log x}$, uniformly in $T$. We also prove a similar results for the logarithmic derivative of the zeta function, and likewise analogous results for the eigenvalues of a random unitary matrix.\n  We use results of this sort to show on the Riemann hypothesis that the av"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.05658","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1502.05658","created_at":"2026-05-18T00:35:18.823824+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.05658v3","created_at":"2026-05-18T00:35:18.823824+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.05658","created_at":"2026-05-18T00:35:18.823824+00:00"},{"alias_kind":"pith_short_12","alias_value":"JPSJQJLSH7AA","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_16","alias_value":"JPSJQJLSH7AAV4MG","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_8","alias_value":"JPSJQJLS","created_at":"2026-05-18T12:29:27.538025+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JPSJQJLSH7AAV4MGV2H4AW4EVO","json":"https://pith.science/pith/JPSJQJLSH7AAV4MGV2H4AW4EVO.json","graph_json":"https://pith.science/api/pith-number/JPSJQJLSH7AAV4MGV2H4AW4EVO/graph.json","events_json":"https://pith.science/api/pith-number/JPSJQJLSH7AAV4MGV2H4AW4EVO/events.json","paper":"https://pith.science/paper/JPSJQJLS"},"agent_actions":{"view_html":"https://pith.science/pith/JPSJQJLSH7AAV4MGV2H4AW4EVO","download_json":"https://pith.science/pith/JPSJQJLSH7AAV4MGV2H4AW4EVO.json","view_paper":"https://pith.science/paper/JPSJQJLS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.05658&json=true","fetch_graph":"https://pith.science/api/pith-number/JPSJQJLSH7AAV4MGV2H4AW4EVO/graph.json","fetch_events":"https://pith.science/api/pith-number/JPSJQJLSH7AAV4MGV2H4AW4EVO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JPSJQJLSH7AAV4MGV2H4AW4EVO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JPSJQJLSH7AAV4MGV2H4AW4EVO/action/storage_attestation","attest_author":"https://pith.science/pith/JPSJQJLSH7AAV4MGV2H4AW4EVO/action/author_attestation","sign_citation":"https://pith.science/pith/JPSJQJLSH7AAV4MGV2H4AW4EVO/action/citation_signature","submit_replication":"https://pith.science/pith/JPSJQJLSH7AAV4MGV2H4AW4EVO/action/replication_record"}},"created_at":"2026-05-18T00:35:18.823824+00:00","updated_at":"2026-05-18T00:35:18.823824+00:00"}