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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 $\\pi(x)$."},"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":"2604.21946","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2026-04-22T05:56:29Z","cross_cats_sorted":[],"title_canon_sha256":"95e2fb6dd7b387d52691412831bab0ee8fc58d2a9e5e202edc1b6411b1e4446b","abstract_canon_sha256":"a22136441ed90d78d06e0f6a49fda2b73357ec8443445f4ab3629d80b12d3949"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:53.410278Z","signature_b64":"fC1Avq/yw9gbBOWca0efMbb3kFrcKAvpaCaoWnDWC3eD052lVVGoW4jfwhkDlZXwJFy1WAjVnsHbBAoiGWdZDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"570fb256fcadb60a600702812919257a8f6119e8bc5061455c30bb983fbae380","last_reissued_at":"2026-06-02T02:04:53.409875Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:53.409875Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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