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We show that the counting function of this set of integers is asymptotic to $K x(\\log\\log x)^\\ell/(\\log x)^{1/(q-1)}$ for explicit constants $K$ and $\\ell$ depending on $q$ and $H$.\n  Second, we con"},"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":"2007.09497","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-07-18T19:10:55Z","cross_cats_sorted":[],"title_canon_sha256":"31da2a236f8f248e5623749b63c38aaabf9e46393b1ac30b9b106547176c3a08","abstract_canon_sha256":"8746502550123e909fbb7a03603f119af2cd2fe7da5a766cf8679145d34ae2b8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:20:25.499649Z","signature_b64":"99kt7qKH7u0TkNTl+E8aquCh23KZ5yXSENs8TTcWCEFfuPYycupx2kKN86TptzL+k5W9jPyxlouLug+6+zn/BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"97a2b802769aa7366a0aa7bf3715b2ee756a414dc552460a02d45df608375641","last_reissued_at":"2026-07-05T01:20:25.499205Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:20:25.499205Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Counting multiplicative groups with prescribed subgroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Greg Martin, Jenna Downey","submitted_at":"2020-07-18T19:10:55Z","abstract_excerpt":"We examine two counting problems that seem very group-theoretic on the surface but, on closer examination, turn out to concern integers with restrictions on their prime factors.\n  First, given an odd prime $q$ and a finite abelian $q$-group $H$, we consider the set of integers $n\\le x$ such that the Sylow $q$-subgroup of the multiplicative group $(\\mathbb Z/n\\mathbb Z)^\\times$ is isomorphic to $H$. 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