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Under mild assumptions on the spectral function $\\psi_\\rho$ associated with $\\rho$, we prove that the expectation of the number $N_n([0,2\\pi])$ of real roots of $f_n$ in the interval $[0,2\\pi]$ satisfies \\[ \\lim_{n \\to +\\infty} \\frac{\\mathbb E\\left [N_n([0,2\\pi])\\right]}{n} = \\frac{2}{\\sqrt{3}}. \\] The latter result "},"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":"1706.01654","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-06-06T08:33:08Z","cross_cats_sorted":[],"title_canon_sha256":"0338c28fd2aee22c3192e0eb0567bad615dd9ccb21896a3e97188a8577aea551","abstract_canon_sha256":"80682da1b4dd251e7d2d7445a32a002a928cbaf3f51df06a51c986c7c3a81529"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:56.359942Z","signature_b64":"9riEYellG0v9bbR5//LmDBaIM3pOITUdpanN9bqd1CDTRQAlrOO1U/9GT9q3cNfWEqEoqZuRqYMaOcf8M/jwDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"535e0b86ee400c546c77cd76b04e91042455accc40d76c982ce132eead55ef4e","last_reissued_at":"2026-05-18T00:42:56.359310Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:56.359310Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the real zeros of random trigonometric polynomials with dependent coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Federico Dalmao, Guillaume Poly, J\\\"urgen Angst","submitted_at":"2017-06-06T08:33:08Z","abstract_excerpt":"We consider random trigonometric polynomials of the form \\[ f_n(t):=\\sum_{1\\le k \\le n} a_{k} \\cos(kt) + b_{k} \\sin(kt), \\] whose entries $(a_{k})_{k\\ge 1}$ and $(b_{k})_{k\\ge 1}$ are given by two independent stationary Gaussian processes with the same correlation function $\\rho$. Under mild assumptions on the spectral function $\\psi_\\rho$ associated with $\\rho$, we prove that the expectation of the number $N_n([0,2\\pi])$ of real roots of $f_n$ in the interval $[0,2\\pi]$ satisfies \\[ \\lim_{n \\to +\\infty} \\frac{\\mathbb E\\left [N_n([0,2\\pi])\\right]}{n} = \\frac{2}{\\sqrt{3}}. \\] The latter result "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.01654","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":"1706.01654","created_at":"2026-05-18T00:42:56.359427+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.01654v1","created_at":"2026-05-18T00:42:56.359427+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.01654","created_at":"2026-05-18T00:42:56.359427+00:00"},{"alias_kind":"pith_short_12","alias_value":"KNPAXBXOIAGF","created_at":"2026-05-18T12:31:24.725408+00:00"},{"alias_kind":"pith_short_16","alias_value":"KNPAXBXOIAGFI3DX","created_at":"2026-05-18T12:31:24.725408+00:00"},{"alias_kind":"pith_short_8","alias_value":"KNPAXBXO","created_at":"2026-05-18T12:31:24.725408+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/KNPAXBXOIAGFI3DXZV3LATURAQ","json":"https://pith.science/pith/KNPAXBXOIAGFI3DXZV3LATURAQ.json","graph_json":"https://pith.science/api/pith-number/KNPAXBXOIAGFI3DXZV3LATURAQ/graph.json","events_json":"https://pith.science/api/pith-number/KNPAXBXOIAGFI3DXZV3LATURAQ/events.json","paper":"https://pith.science/paper/KNPAXBXO"},"agent_actions":{"view_html":"https://pith.science/pith/KNPAXBXOIAGFI3DXZV3LATURAQ","download_json":"https://pith.science/pith/KNPAXBXOIAGFI3DXZV3LATURAQ.json","view_paper":"https://pith.science/paper/KNPAXBXO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.01654&json=true","fetch_graph":"https://pith.science/api/pith-number/KNPAXBXOIAGFI3DXZV3LATURAQ/graph.json","fetch_events":"https://pith.science/api/pith-number/KNPAXBXOIAGFI3DXZV3LATURAQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KNPAXBXOIAGFI3DXZV3LATURAQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KNPAXBXOIAGFI3DXZV3LATURAQ/action/storage_attestation","attest_author":"https://pith.science/pith/KNPAXBXOIAGFI3DXZV3LATURAQ/action/author_attestation","sign_citation":"https://pith.science/pith/KNPAXBXOIAGFI3DXZV3LATURAQ/action/citation_signature","submit_replication":"https://pith.science/pith/KNPAXBXOIAGFI3DXZV3LATURAQ/action/replication_record"}},"created_at":"2026-05-18T00:42:56.359427+00:00","updated_at":"2026-05-18T00:42:56.359427+00:00"}