{"paper":{"title":"Strong orthogonality between the Mobius function and nonlinear exponential functions in short intervals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bingrong Huang","submitted_at":"2014-12-06T13:51:11Z","abstract_excerpt":"Let $\\mu(n)$ be the M\\\"obius function, $e(z) = \\exp(2\\pi iz)$, $x$ real and $2\\leq y \\leq x$. This paper proves two sequences $(\\mu(n))$ and $(e(n^k \\alpha))$ are strongly orthogonal in short intervals. That is, if $k \\geq 3$ being fixed and $y\\geq x^{1-1/4+\\varepsilon}$, then for any $A>0$, we have \\[\n  \\sum_{x< n \\leq x+y} \\mu(n) e\\left(n^k \\alpha \\right) \\ll y(\\log y)^{-A} \\] uniformly for $\\alpha \\in \\mathbb{R}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.2237","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"}