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Specifically we shall be interested in effective exponential bounds of the form \\[ |\\vartheta(x)-x| < a \\;x \\;(\\ln x)^b \\; \\exp\\left(-c\\; \\sqrt{\\ln x}\\right); \\qquad (x \\geq x_0). \\] Herein we shall convert these effective bounds on $\\vartheta(x)$ into effective exponential bounds on the prime gaps $g_n = p_{n+1}-p_n$. Specifically we shall establish a number of effective exponential bounds of the form \\[ {g_n\\over p_n} < { 2a \\;(\\ln p_n)^b \\; \\exp\\left(-c\\; "},"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":"2211.06469","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-11-11T20:08:26Z","cross_cats_sorted":[],"title_canon_sha256":"18d3ccf5cc4380b0ad3a9a1bf05f2b92943b9cfde90b6fc1c6c9cfaa68e23ebb","abstract_canon_sha256":"55691fb63c7355bca3de28d5f6c1d7ad426d553d37d8ad314a6777d4a4d4dba8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:54:29.261038Z","signature_b64":"BEPPK/IWygPRfAf47mSqOIhpV5jie+QDauB8InIQklF0cuX8vel6Uv1aVh8TSNPnbp7uR+NejS8KfMMIrlhlDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0f79b75588bf48877c12a35cf6aa0aca744594eb816211bb6f54b6f3f3f8100","last_reissued_at":"2026-07-05T10:54:29.260489Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:54:29.260489Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Effective exponential bounds on the prime gaps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Matt Visser (Victoria University of Wellington)","submitted_at":"2022-11-11T20:08:26Z","abstract_excerpt":"Over the last 50 years a large number of effective exponential bounds on the first Chebyshev function $\\vartheta(x)$ have been obtained. Specifically we shall be interested in effective exponential bounds of the form \\[ |\\vartheta(x)-x| < a \\;x \\;(\\ln x)^b \\; \\exp\\left(-c\\; \\sqrt{\\ln x}\\right); \\qquad (x \\geq x_0). \\] Herein we shall convert these effective bounds on $\\vartheta(x)$ into effective exponential bounds on the prime gaps $g_n = p_{n+1}-p_n$. 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