{"paper":{"title":"M\\\"obius formulas for densities of sets of prime ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ashwin Sah, Michael Kural, Vaughan McDonald","submitted_at":"2019-07-05T16:25:25Z","abstract_excerpt":"We generalize results of Alladi, Dawsey, and Sweeting and Woo for Chebotarev densities to general densities of sets of primes. We show that if $K$ is a number field and $S$ is any set of prime ideals with natural density $\\delta(S)$ within the primes, then \\[ -\\lim_{X \\to \\infty}\\sum_{\\substack{2 \\le \\operatorname{N}(\\mathfrak{a})\\le X\\\\ \\mathfrak{a} \\in D(K,S)}}\\frac{\\mu(\\mathfrak{a})}{\\operatorname{N}(\\mathfrak{a})} = \\delta(S), \\] where $\\mu(\\mathfrak{a})$ is the generalized M\\\"obius function and $D(K,S)$ is the set of integral ideals $ \\mathfrak{a} \\subseteq \\mathcal{O}_K$ with unique prim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.02914","kind":"arxiv","version":2},"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"}