{"paper":{"title":"On the Sign Changes of a Weighted Divisor Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Lirui Jia, Tianxin Cai, Wenguang Zhai","submitted_at":"2016-03-16T06:44:20Z","abstract_excerpt":"Let $S\\big(x; \\frac{a_1}{q_1}, \\frac{a_2}{q_2}\\big)=\\mathop{{\\sum}'}_{mn\\leq x} \\cos\\big(2\\pi m\\frac{a_1}{q_1}\\big)\\sin\\big(2\\pi n\\frac{a_2}{q_2}\\big)$ with $x\\geq q_1q_2, 1\\leq a_i\\leq q_i$, and $(a_i, q_i)=1$ ($i=1, 2$). We study the sign changes of $S\\big(x; \\frac{a_1}{q_1}, \\frac{a_2}{q_2}\\big)$, and prove that for a sufficiently large constant $C$, $S\\big(x; \\frac{a_1}{q_1}, \\frac{a_2}{q_2}\\big)$ changes sign in the interval $[T,T+C\\sqrt{T}]$ for any large $T$. Meanwhile, we show that for a small constant $c'$, there exist infinitely many subintervals of length $c'\\sqrt{T}\\log^{-7}T$ in $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.04977","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"}