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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 $\\pi(x;q,a)$ "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1802.00085","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-01-31T22:13:20Z","cross_cats_sorted":[],"title_canon_sha256":"f384677eb4928989ed8d769cbc525e6825b26afca773079f5012719747212dd9","abstract_canon_sha256":"6243140cd25c25d7cb9896a04a45fbfd6173f410cc6b6d032d7019f0a0da0255"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:44.704953Z","signature_b64":"CAtYhDSnCdhKWrUZM02tmxjCp64CRLGov2LBYxa2cJx8qR2+Li0VKpXbsL3bCYF1hiyUcyMuyuKmYbuNFwT2DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70fb53d50e2f7f70485b276b82d2cf1c249d3788d83094ff6b4eb75806e81888","last_reissued_at":"2026-05-17T23:59:44.704597Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:44.704597Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $q$). 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