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We prove that $\\mathbb{E} N[0,1-\\epsilon] \\sim \\frac{\\sqrt{\\gamma}}{2\\pi} |\\log \\epsilon|$ as $\\epsilon \\downarrow 0$, where $N[0,r]$ denotes the number of real zeroes of $f$ in the interval $[0,r]$."},"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":"1709.02937","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-09-09T09:54:53Z","cross_cats_sorted":[],"title_canon_sha256":"ebea1fd62df750a4bd0bcbf9edcee65c5adccb78eb44d9cc940d7514d7db6a78","abstract_canon_sha256":"8f5fee1ea31a1bfa108e376d0b50f931ba33de8202e04ef152740fe06d4cca61"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:43.212196Z","signature_b64":"lAWbpenXmypbgGDnNd2G6vrP0kd28wX+GhnsI+RVL2CWFpKe1xaYgdxePacHg0sMirJhktHmwMy15P52sRAyAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d87d1ec3fd2795d2010b61c68f4be71b62c2a18e40ee9bdf5cb5e3fcdc260df","last_reissued_at":"2026-05-18T00:33:43.211574Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:43.211574Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Expected number of real zeros of random Taylor Series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hendrik Flasche, Zakhar Kabluchko","submitted_at":"2017-09-09T09:54:53Z","abstract_excerpt":"Let $\\xi_0,\\xi_1,\\ldots$ be i.i.d. random variables with zero mean and unit variance. Consider a random Taylor series of the form $f(z)=\\sum_{k=0}^\\infty \\xi_k c_k z^k$, where $c_0,c_1,\\ldots$ is a real sequence such that $c_n^2$ is regularly varying with index $\\gamma-1$, where $\\gamma>0$. We prove that $\\mathbb{E} N[0,1-\\epsilon] \\sim \\frac{\\sqrt{\\gamma}}{2\\pi} |\\log \\epsilon|$ as $\\epsilon \\downarrow 0$, where $N[0,r]$ denotes the number of real zeroes of $f$ in the interval $[0,r]$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.02937","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":""},"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":"1709.02937","created_at":"2026-05-18T00:33:43.211665+00:00"},{"alias_kind":"arxiv_version","alias_value":"1709.02937v2","created_at":"2026-05-18T00:33:43.211665+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.02937","created_at":"2026-05-18T00:33:43.211665+00:00"},{"alias_kind":"pith_short_12","alias_value":"TWD5D3B72J4V","created_at":"2026-05-18T12:31:46.661854+00:00"},{"alias_kind":"pith_short_16","alias_value":"TWD5D3B72J4V2IAQ","created_at":"2026-05-18T12:31:46.661854+00:00"},{"alias_kind":"pith_short_8","alias_value":"TWD5D3B7","created_at":"2026-05-18T12:31:46.661854+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/TWD5D3B72J4V2IAQWYOGR5F6OG","json":"https://pith.science/pith/TWD5D3B72J4V2IAQWYOGR5F6OG.json","graph_json":"https://pith.science/api/pith-number/TWD5D3B72J4V2IAQWYOGR5F6OG/graph.json","events_json":"https://pith.science/api/pith-number/TWD5D3B72J4V2IAQWYOGR5F6OG/events.json","paper":"https://pith.science/paper/TWD5D3B7"},"agent_actions":{"view_html":"https://pith.science/pith/TWD5D3B72J4V2IAQWYOGR5F6OG","download_json":"https://pith.science/pith/TWD5D3B72J4V2IAQWYOGR5F6OG.json","view_paper":"https://pith.science/paper/TWD5D3B7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1709.02937&json=true","fetch_graph":"https://pith.science/api/pith-number/TWD5D3B72J4V2IAQWYOGR5F6OG/graph.json","fetch_events":"https://pith.science/api/pith-number/TWD5D3B72J4V2IAQWYOGR5F6OG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TWD5D3B72J4V2IAQWYOGR5F6OG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TWD5D3B72J4V2IAQWYOGR5F6OG/action/storage_attestation","attest_author":"https://pith.science/pith/TWD5D3B72J4V2IAQWYOGR5F6OG/action/author_attestation","sign_citation":"https://pith.science/pith/TWD5D3B72J4V2IAQWYOGR5F6OG/action/citation_signature","submit_replication":"https://pith.science/pith/TWD5D3B72J4V2IAQWYOGR5F6OG/action/replication_record"}},"created_at":"2026-05-18T00:33:43.211665+00:00","updated_at":"2026-05-18T00:33:43.211665+00:00"}