{"paper":{"title":"Euler's Function on Products of Primes in Progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Amir Akbary, Forrest J. Francis","submitted_at":"2018-10-26T20:39:30Z","abstract_excerpt":"We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $\\varphi(n)$ and the Riemann Hypothesis. Among other things, we prove that for $1\\leq q\\leq 10$ and for $q=12, 14$, the generalized Riemann Hypothesis for the Dedekind zeta function of the cyclotomic field $\\mathbb{Q}(e^{2\\pi i/q})$ is true if and only if for all integers $k\\geq 1$ we have \\[\\frac{\\bar{N}_k}{\\varphi(\\bar{N}_k)(\\log(\\varphi(q)\\log{\\bar{N}_k}))^{\\frac{1}{\\varphi(q)}}} > \\frac{1}{C(q,1)}.\\] Here $\\bar{N}_k$ is the product of the first $k$ primes in the ar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.11524","kind":"arxiv","version":1},"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"}