{"paper":{"title":"Ergodic theorems with arithmetical weights","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Christophe Cuny, Michel Weber","submitted_at":"2014-12-24T10:46:16Z","abstract_excerpt":"We prove that the divisor function $d(n)$ counting the number of divisors of the integer $n$, is a good weighting function for the pointwise ergodic theorem. For any measurable dynamical system $(X, {\\mathcal A},\\nu,\\tau)$ and any $f\\in L^p(\\nu)$, $p>1$, the limit $$ \\lim_{n\\to \\infty}{1\\over \\sum_{k=1}^{n} d(k)} \\sum_{k=1}^{n} d(k)f(\\tau^k x)$$ exists $\\nu$-almost everywhere. We also obtain similar results for other arithmetical functions, like $\\theta(n)$ function counting the number of squarefree divisors of $n$ and the generalized Euler totient function $J_s(n)$, $s>0$. We use Bourgain's m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.7640","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"}