{"paper":{"title":"A Generalization of a Result of Hardy and Littlewood","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ilya Vinogradov","submitted_at":"2007-09-18T16:53:24Z","abstract_excerpt":"In this note we study the growth of \\sum_{m=1}^M\\frac1{\\|m\\alpha\\|} as a function of M for different classes of \\alpha\\in[0,1). Hardy and Littlewood showed that for numbers of bounded type, the sum is \\simeq M\\log M. We give a very simple proof for it. Further we show the following for generic \\alpha. For a non-decreasing function \\phi tending to infinity, \\limsup_{M\\to\\infty}\\frac1{\\phi(\\log M)}\\bigg[\\frac1{M\\log M}\\sum_{m=1}^M\\frac1{\\|m\\alpha\\|}\\bigg] is zero or infinity according as \\sum\\frac1{k\\phi(k)} converges or diverges."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0709.2882","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"}