{"paper":{"title":"The $S$-$E$ route to the Chebyshev bounds for the prime-counting function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"An order-of-magnitude bound on the weighted prime sum S(x) implies the Chebyshev bounds for the prime-counting function.","cross_cats":[],"primary_cat":"math.GM","authors_text":"Kai Hubbard","submitted_at":"2026-04-22T05:56:29Z","abstract_excerpt":"The Chebyshev bounds for the prime-counting function, i.e., $\\pi(x) \\asymp x/\\ln x$, is established in a new way. This new approach follows the outline $\\pi(x) \\asymp \\bigl(\\sum_{p \\le x} \\sqrt{\\ln p / p}\\bigr)^{2} \\asymp x/\\ln x$. Here, the second $\\asymp$ is derived from the classical estimate by Mertens, i.e., $\\sum_{p \\le x} (\\ln p)/p = \\ln x + O(1)$; while the first $\\asymp$ is proved by considering the difference $\\bigl(\\sum_{p \\le x} \\sqrt{\\ln p/p}\\bigr)^{2} - \\sum_{p \\le x} (\\ln p)/p$, which is shown as having the same order as $\\pi(x)$."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that the order-of-magnitude estimate S(x) ≍ sqrt(x / log x) implies the Chebyshev bounds π(x) ≍ x / log x through a short and transparent chain of inequalities. The mechanism passes through E(x), which we show satisfies E(x) ≍ π(x) whenever the size estimate for S(x) holds.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the specific inequalities relating E(x) to π(x) hold with the claimed constants and without hidden restrictions when S(x) satisfies the given order-of-magnitude bound.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"S(x) ≍ sqrt(x / log x) implies π(x) ≍ x / log x because E(x) ≍ π(x) under that assumption, with the S estimate following from Mertens' theorem.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"An order-of-magnitude bound on the weighted prime sum S(x) implies the Chebyshev bounds for the prime-counting function.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"dea6b7b8a594d126d5b4649621ab700a260ed9a8229cad4266acc7a7b8295693"},"source":{"id":"2604.21946","kind":"arxiv","version":2},"verdict":{"id":"eb9e873d-8f35-4489-9502-ba2c344fa9f5","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-09T23:14:37.610783Z","strongest_claim":"We prove that the order-of-magnitude estimate S(x) ≍ sqrt(x / log x) implies the Chebyshev bounds π(x) ≍ x / log x through a short and transparent chain of inequalities. The mechanism passes through E(x), which we show satisfies E(x) ≍ π(x) whenever the size estimate for S(x) holds.","one_line_summary":"S(x) ≍ sqrt(x / log x) implies π(x) ≍ x / log x because E(x) ≍ π(x) under that assumption, with the S estimate following from Mertens' theorem.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the specific inequalities relating E(x) to π(x) hold with the claimed constants and without hidden restrictions when S(x) satisfies the given order-of-magnitude bound.","pith_extraction_headline":"An order-of-magnitude bound on the weighted prime sum S(x) implies the Chebyshev bounds for the prime-counting function."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.21946/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T15:34:32.522017Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-20T02:12:51.764853Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"618276ecb516c52429d497b4a1c3fe97e549413b57ad07164e788189b675eaae"},"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"}