{"paper":{"title":"G\\'al-type GCD sums beyond the critical line","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andriy Bondarenko, Kristian Seip, Titus Hilberdink","submitted_at":"2015-12-11T19:04:09Z","abstract_excerpt":"We prove that \\[ \\sum_{k,{\\ell}=1}^N\\frac{(n_k,n_{\\ell})^{2\\alpha}}{(n_k n_{\\ell})^{\\alpha}} \\ll N^{2-2\\alpha} (\\log N)^{b(\\alpha)} \\] holds for arbitrary integers $1\\le n_1<\\cdots < n_N$ and $0<\\alpha<1/2$ and show by an example that this bound is optimal, up to the precise value of the exponent $b(\\alpha)$. This estimate complements recent results for $1/2\\le \\alpha \\le 1$ and shows that there is no \"trace\" of the functional equation for the Riemann zeta function in estimates for such GCD sums when $0<\\alpha<1/2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03758","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"}