{"paper":{"title":"GCD sums from Poisson integrals and systems of dilated functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CV","math.FA"],"primary_cat":"math.NT","authors_text":"Christoph Aistleitner, Istvan Berkes, Kristian Seip","submitted_at":"2012-10-02T12:10:25Z","abstract_excerpt":"Upper bounds for GCD sums of the form [\\sum_{k,{\\ell}=1}^N\\frac{(\\gcd(n_k,n_{\\ell}))^{2\\alpha}}{(n_k n_{\\ell})^\\alpha}] are proved, where $(n_k)_{1 \\leq k \\leq N}$ is any sequence of distinct positive integers and $0<\\alpha \\le 1$; the estimate for $\\alpha=1/2$ solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for $\\alpha=1/2$. The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.0741","kind":"arxiv","version":5},"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"}