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It is shown that \\[ (\\sum_{n=1}^{N} |a_n|^2|\\mu(n)|[d(n)]^{\\frac{\\log q}{\\log 2} -1})^{1/2}\\le \\| F\\|_q \\] when $0<q<2$; here $\\mu$ is the M\\\"{o}bius function and $d$ the divisor function. This result is used to prove that the $L^q$ norm of $D_N(s):=\\sum_{n=1}^{N} n^{-1/2-s}$ satisfies $\\|D_N\\|_q\\gg (\\log N)^{q/4}$ for $0<q<\\infty$. By Helson's generalization of the M. Riesz theorem on the conjugation operator, the re"},"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":"1405.6516","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-05-26T09:32:37Z","cross_cats_sorted":["math.CV","math.FA"],"title_canon_sha256":"d261a028ccf9a830b4c45cd1ce34148d3fc990dded8aeaef4737bae07bb138e9","abstract_canon_sha256":"7d4195eb6be9b8842c570a0f9ab2ad820550c567a6b8c75b9746c4009766baba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:20.802483Z","signature_b64":"7y9O4OvCF5em+lgPcAun8f++WECq40rjCwdnCNqnrdUvj6sQ9XVSvdCKzqWUUySU8JXZ3Wu02WDx+jcw1oWhDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70050da5964a7f4c521a09ef701c51bc17060bb3fceb54c3f07edcdd376f1604","last_reissued_at":"2026-05-18T02:28:20.801940Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:20.801940Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An inequality of Hardy--Littlewood type for Dirichlet polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.FA"],"primary_cat":"math.NT","authors_text":"Andriy Bondarenko, Kristian Seip, Winston Heap","submitted_at":"2014-05-26T09:32:37Z","abstract_excerpt":"The $L^q$ norm of a Dirichlet polynomial $F(s)=\\sum_{n=1}^{N} a_n n^{-s}$ is defined as \\[\\| F\\|_q:=(\\lim_{T\\to\\infty}\\frac{1}{T}\\int_{0}^T |F(it)|^qdt)^{1/q}\\] for $0<q<\\infty$. It is shown that \\[ (\\sum_{n=1}^{N} |a_n|^2|\\mu(n)|[d(n)]^{\\frac{\\log q}{\\log 2} -1})^{1/2}\\le \\| F\\|_q \\] when $0<q<2$; here $\\mu$ is the M\\\"{o}bius function and $d$ the divisor function. This result is used to prove that the $L^q$ norm of $D_N(s):=\\sum_{n=1}^{N} n^{-1/2-s}$ satisfies $\\|D_N\\|_q\\gg (\\log N)^{q/4}$ for $0<q<\\infty$. By Helson's generalization of the M. 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