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This improves a result of Goldston and Gonek by a factor of 2. The first method consists in bounding the auxiliary function $S_1(t) = \\int_0^{t} S(u) du$ using extremal functions constructed by Carneiro, Littmann and Vaaler. We then relate the size of $S(t)$ to the size of the functions $S_1(t\\pm h)-S_1(t)$ when $h\\asymp 1/\\lo"},"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":"1309.1526","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-06T03:10:26Z","cross_cats_sorted":[],"title_canon_sha256":"7c04b6d7aa2c00a10a63d4843643ceb0ef5c3d88d62516eeb719c95f661f5cc3","abstract_canon_sha256":"9ba8221bec39b7d5324506a9ed082afe1a266960773cd3a8ea05b15f3bfe7a72"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:59.228581Z","signature_b64":"SzN4X2qC3kvM31FVdXJ+NyJHrXunpc6qrxHNVdr4BD/sAYiXNB8tNR6txoc1nLiJPQxC73YpyPMv89+Tkb48Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eba57447d0772a3854a130efb179789538dbb015d9ff411b52d2d2da89fcbddf","last_reissued_at":"2026-05-18T03:13:59.227919Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:59.227919Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bounding $S(t)$ and $S_1(t)$ on the Riemann hypothesis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Emanuel Carneiro, Micah B. Milinovich, Vorrapan Chandee","submitted_at":"2013-09-06T03:10:26Z","abstract_excerpt":"Let $\\pi S(t)$ denote the argument of the Riemann zeta-function, $\\zeta(s)$, at the point $s=\\frac{1}{2}+it$. Assuming the Riemann hypothesis, we present two proofs of the bound $$ |S(t)| \\leq \\left(\\tfrac{1}{4} + o(1) \\right)\\tfrac{\\log t}{\\log \\log t} $$ for large $t$. This improves a result of Goldston and Gonek by a factor of 2. The first method consists in bounding the auxiliary function $S_1(t) = \\int_0^{t} S(u) du$ using extremal functions constructed by Carneiro, Littmann and Vaaler. 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