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Our results improve an estimate of Bhargava-Shankar-Wang in \\cite{bhargava2022squarefree}. In particular we show the following:\n  $\\bullet$ When $n\\ge 4,$ the number of rings of rank $n$ over $\\mathbb{Z}$ with discriminant less than or equal to $X$ is $$\\"},"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":"2410.12970","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-10-16T19:04:26Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"519f52cc4eb508f6e7d048751d1593f33d2518699761dad2d82ef0f4b287aed7","abstract_canon_sha256":"0bfd358fcd596c876e98ac73c6a8e87f4f5a77a631d1bf0997773d68773fa073"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-10T01:09:13.464417Z","signature_b64":"7SW/c0iN+qilYojISWlVz8xhzyahUJE0nizcfsJrETDw3RAFVQL/s2SrgikGxV7Wxe2jQDI+4rHJkcMGoDhhBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b591970ad535bcab4bd81ad1a56fa5e5cfaef5059e31fd97f8627a9a96c6434","last_reissued_at":"2026-06-10T01:09:13.463816Z","signature_status":"signed_v1","first_computed_at":"2026-06-10T01:09:13.463816Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weakly Divisible Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.NT","authors_text":"Gaurav Digambar Patil","submitted_at":"2024-10-16T19:04:26Z","abstract_excerpt":"We define a new class of rings parameterized by binary forms of a certain type, and give an effective lower bound for the number of such rings whose discriminant is less than a bound $X$. We also obtain a lower bound for the number of number fields whose ring of integers lies in the above class and whose discriminant is less than a bound $X$. Our results improve an estimate of Bhargava-Shankar-Wang in \\cite{bhargava2022squarefree}. 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