{"paper":{"title":"Explicit smoothed prime ideals theorems under GRH","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Giuseppe Molteni, Lo\\\"ic Greni\\'e","submitted_at":"2013-12-16T19:10:48Z","abstract_excerpt":"Let $\\psi_{\\mathbb K}$ be the Chebyshev function of a number field $\\mathbb K$. Let $\\psi^{(1)}_{\\mathbb K}(x):=\\int_{0}^{x}\\psi_{\\mathbb K}(t)\\,d t$ and $\\psi^{(2)}_{\\mathbb K}(x):=2\\int_{0}^{x}\\psi^{(1)}_{\\mathbb K}(t)\\,d t$. We prove under GRH explicit inequalities for the differences $|\\psi^{(1)}_{\\mathbb K}(x) - \\tfrac{x^2}{2}|$ and $|\\psi^{(2)}_{\\mathbb K}(x) - \\tfrac{x^3}{3}|$. We deduce an efficient algorithm for the computation of the residue of the Dedekind zeta function and a bound on small-norm prime ideals."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.4465","kind":"arxiv","version":5},"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"}