{"paper":{"title":"Explicit formulae for primes in arithmetic progressions, I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tomohiro Yamada","submitted_at":"2013-06-22T13:27:17Z","abstract_excerpt":"We shall give an explicit formula for $\\psi(x, q, a)$ with an error term of the form $C/\\log^\\alpha x$ under the condition that $q<\\log^{\\alpha_1} x$ is nonexceptional, for various values of $\\alpha$ and $\\alpha_1$. We shall also give an explicit formula for $\\psi(x, q, a)$ with error terms $C/\\log^A x$ working whether $q$ is exceptional or nonexceptional, but under the condition that $\\frac{0.4923A}{\\pi}q^{1/2}\\log^2 q<\\log x/\\log\\log x$. Moreover, we shall give an explicit form of Bombieri-Vinogradov theorem over non-exceptional moduli."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.5322","kind":"arxiv","version":4},"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"}