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For $S_n$-fields, $n=3,4,5$, we give a different proof which depends on the strong Artin conjecture. Let $N_{K,C}$ be the smallest prime for which Frob$_p$ belongs to $C$. For $S_3$-fields, we obtain an unconditional result for the average of $N_{K,C}$ for $C=[(12)]$."},"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":"1601.03012","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-12T19:50:59Z","cross_cats_sorted":[],"title_canon_sha256":"b0e9556ded9616d5964146029ab178ea512d53d341b65c72ec9d2a48920868a3","abstract_canon_sha256":"3c0b258b440c0740bdd69288637c33cdb25c8eb1e995978a1eba404e5e47d050"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:59.111585Z","signature_b64":"/zoLigfHH/gk4rcnGglDhB07Soqxnv/m50d7Lx/gyVniVscAiOpaZrdyZ/AdUJX/lk7R2g1rX+9IT4gLOcyiCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41640f3ee7385ebf6aadd2d7c8cfe740951a68e3eff0373a56425620ac826606","last_reissued_at":"2026-05-18T01:22:59.110978Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:59.110978Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The average of the smallest prime in a conjugacy class","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Henry H. Kim, Peter J. Cho","submitted_at":"2016-01-12T19:50:59Z","abstract_excerpt":"Let $C$ be a conjugacy class of $S_n$ and $K$ an $S_n$-field. Let $n_{K,C}$ be the smallest prime which is ramified or whose Frobenius automorphism Frob$_p$ does not belong to $C$. Under some technical conjectures, we compute the average of $n_{K,C}$. For $S_3$ and $S_4$-fields, our result is unconditional. For $S_n$-fields, $n=3,4,5$, we give a different proof which depends on the strong Artin conjecture. Let $N_{K,C}$ be the smallest prime for which Frob$_p$ belongs to $C$. 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