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In the region $0 < a < \\frac{-\\sigma}{2\\pi e}$ the zeroes are asymptotically located at the lines $\\sigma + 4a + 2m =0$ with integer $m$. If $N(p)$ is the number of real zeroes of $\\zeta(-p,a)$ with given $p$ then $$\\lim_{p\\to\\infty}\\frac{N(p)}{p}=\\frac{1}{\\pi e}.$$ As a corollary we have a"},"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":"math/0205183","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GM","submitted_at":"2002-05-16T10:37:25Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"f145c21e837d604bea7e653fe6b6e5ae913c6de1a21c44b2cfe9f55f88ffc1aa","abstract_canon_sha256":"12b719ef6e34930ec4a5e4080deaa07f266fdd72daa40e784459ec78ebcb681b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:29.560972Z","signature_b64":"dUhWggez1FBNJADs3rRTaGTeBdEeNcQpThwh56+PX3+9fb5ztm1PBOxOAvFOo0ev0AIAF7t9yzjYbHFy6q/lAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c24c006c982842937a3e7f61a82dd7a0bfa34fd89b8137bffe647b043a4fcd9","last_reissued_at":"2026-05-18T01:05:29.560570Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:29.560570Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the real zeroes of the Hurwitz zeta-function and Bernoulli polynomials","license":"","headline":"","cross_cats":["math.CA"],"primary_cat":"math.GM","authors_text":"A.P. Veselov, J.P. Ward","submitted_at":"2002-05-16T10:37:25Z","abstract_excerpt":"The behaviour of real zeroes of the Hurwitz zeta function $$\\zeta (s,a)=\\sum_{r=0}^{\\infty}(a+r)^{-s}\\qquad\\qquad a > 0$$ is investigated. It is shown that $\\zeta (s,a)$ has no real zeroes $(s=\\sigma,a)$ in the region $a >\\frac{-\\sigma}{2\\pi e}+\\frac{1}{4\\pi e}\\log (-\\sigma) +1$ for large negative $\\sigma$. In the region $0 < a < \\frac{-\\sigma}{2\\pi e}$ the zeroes are asymptotically located at the lines $\\sigma + 4a + 2m =0$ with integer $m$. If $N(p)$ is the number of real zeroes of $\\zeta(-p,a)$ with given $p$ then $$\\lim_{p\\to\\infty}\\frac{N(p)}{p}=\\frac{1}{\\pi e}.$$ As a corollary we have a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0205183","kind":"arxiv","version":1},"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":"math/0205183","created_at":"2026-05-18T01:05:29.560631+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0205183v1","created_at":"2026-05-18T01:05:29.560631+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0205183","created_at":"2026-05-18T01:05:29.560631+00:00"},{"alias_kind":"pith_short_12","alias_value":"PQSMABWJQKCC","created_at":"2026-05-18T12:25:51.375804+00:00"},{"alias_kind":"pith_short_16","alias_value":"PQSMABWJQKCCSN5D","created_at":"2026-05-18T12:25:51.375804+00:00"},{"alias_kind":"pith_short_8","alias_value":"PQSMABWJ","created_at":"2026-05-18T12:25:51.375804+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/PQSMABWJQKCCSN5D473BVAW5PI","json":"https://pith.science/pith/PQSMABWJQKCCSN5D473BVAW5PI.json","graph_json":"https://pith.science/api/pith-number/PQSMABWJQKCCSN5D473BVAW5PI/graph.json","events_json":"https://pith.science/api/pith-number/PQSMABWJQKCCSN5D473BVAW5PI/events.json","paper":"https://pith.science/paper/PQSMABWJ"},"agent_actions":{"view_html":"https://pith.science/pith/PQSMABWJQKCCSN5D473BVAW5PI","download_json":"https://pith.science/pith/PQSMABWJQKCCSN5D473BVAW5PI.json","view_paper":"https://pith.science/paper/PQSMABWJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0205183&json=true","fetch_graph":"https://pith.science/api/pith-number/PQSMABWJQKCCSN5D473BVAW5PI/graph.json","fetch_events":"https://pith.science/api/pith-number/PQSMABWJQKCCSN5D473BVAW5PI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PQSMABWJQKCCSN5D473BVAW5PI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PQSMABWJQKCCSN5D473BVAW5PI/action/storage_attestation","attest_author":"https://pith.science/pith/PQSMABWJQKCCSN5D473BVAW5PI/action/author_attestation","sign_citation":"https://pith.science/pith/PQSMABWJQKCCSN5D473BVAW5PI/action/citation_signature","submit_replication":"https://pith.science/pith/PQSMABWJQKCCSN5D473BVAW5PI/action/replication_record"}},"created_at":"2026-05-18T01:05:29.560631+00:00","updated_at":"2026-05-18T01:05:29.560631+00:00"}