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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1412.7640","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-12-24T10:46:16Z","cross_cats_sorted":[],"title_canon_sha256":"2a95b1b08a82cd70805c359d805543cbb45c6ec3088643b8b737c4ee6091e882","abstract_canon_sha256":"f6c2e770e54748c91ec741d4d845f535dab655e88b03890a4b8b305344170445"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:58.698798Z","signature_b64":"vImWABXRr+ceTu8ORmepl1XU6FETbojxiigYJqaE52Az/xhkhZuFsZf7WCWmQQIMeZOeSLD0fX8T+7a17p22DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7cf62f96b2e4836647b8561456989a24c561944f16cce44e7d3efcda9cec3210","last_reissued_at":"2026-05-18T00:39:58.698168Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:58.698168Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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