{"paper":{"title":"Explicit bounds for primes in arithmetic progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrew Rechnitzer, Greg Martin, Kevin O'Bryant, Michael A. Bennett","submitted_at":"2018-01-31T22:13:20Z","abstract_excerpt":"We derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\\gcd(a,q)=1$ and $3 \\leq q \\leq 10^5$, and $\\theta(x;q,a)$ denotes the sum of the logarithms of the primes $p \\equiv a \\pmod{q}$ with $p \\leq x$, we show that $$ \\bigg| \\theta (x; q, a) - \\frac{x}{\\phi (q)} \\bigg| < \\frac1{160} \\frac{x}{\\log x}, $$ for all $x \\ge 8 \\cdot 10^9$ (with sharper constants obtained for individual such moduli $q$). We establish inequalities of the same shape for the other standard prime-counting functions $\\pi(x;q,a)$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.00085","kind":"arxiv","version":3},"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"}