{"paper":{"title":"Ergodic averages with prime divisor weights in $L^{1}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.NT"],"primary_cat":"math.DS","authors_text":"Zoltan Buczolich","submitted_at":"2016-10-03T12:19:36Z","abstract_excerpt":"We show that $ { \\omega }(n)$ and $ { \\Omega }(n)$, the number of distinct prime factors of $n$ and the number of distinct prime factors of $n$ counted according to multiplicity are good weighting functions for the pointwise ergodic theorem in $L^{1}$. That is, if $g$ denotes one of these functions and $S_{g,K}=\\sum_{n\\leq K}g(n)$ then for every ergodic dynamical system $(X, { { \\cal A } },\\mu, { \\tau })$ and every $f\\in L^{1}(X)$ $$\\lim_{K\\to { \\infty }} \\frac{1}{S_{g,K}}\\sum_{n=1}^{K} g(n)f( { \\tau }^{n}x)=\\int_{X}fd\\mu \\text{ for $\\mu$ a.e. }x\\in X. $$\n  This answers a question raised by C."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.00511","kind":"arxiv","version":1},"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"}