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These products can be expressed using $\\prod_{k=1}^n \\sin\\Big(\\frac{k\\theta}{2}\\Big)$ and $\\prod_{k=1}^n \\cos\\Big(\\frac{k\\theta}{2}\\Big)$ respectively. We prove an estimate for $P_n$ at a point near where its maximum occurs. Finally, we give an asymptotic formula for the maximum of the Fourier coefficients of $Q_n$."},"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":"1210.7380","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-10-27T22:47:53Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"4f28c32cf8a0b15e848277c5dbaa520dcf7d17a15a766c674f85df429090989f","abstract_canon_sha256":"cf683513f80b196062d69e91c313a1c9cf0719434e6a07458bdc9406a892fb9b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:42:10.242072Z","signature_b64":"XYXOtj6S1EsFGfZt/pbOSI6BNKQ5tP3lzM84HGhhUTUWOI41fe6slGjqVSkgfjLpeNn+YQy5ga3T27OcqihYDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"12bbdf9c37e751bb32ceeb14d0e8c7784fd61ed721c03b69366fb5aca0894991","last_reissued_at":"2026-05-18T03:42:10.241158Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:42:10.241158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Estimates for the norms of products of sines and cosines","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CA","authors_text":"Jordan Bell","submitted_at":"2012-10-27T22:47:53Z","abstract_excerpt":"In this paper we prove asymptotic formulas for the $L^p$ norms of $P_n(\\theta)=\\prod_{k=1}^n (1-e^{ik\\theta})$ and $Q_n(\\theta)=\\prod_{k=1}^n (1+e^{ik\\theta})$. These products can be expressed using $\\prod_{k=1}^n \\sin\\Big(\\frac{k\\theta}{2}\\Big)$ and $\\prod_{k=1}^n \\cos\\Big(\\frac{k\\theta}{2}\\Big)$ respectively. We prove an estimate for $P_n$ at a point near where its maximum occurs. 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