{"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. Riesz theorem on the conjugation operator, the re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.6516","kind":"arxiv","version":3},"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"}