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We show that the error due to truncation is bounded in absolute value by, and of the same sign as, the first neglected term for all nonnegative $n$. Consequently, we obtain optimal sharp bounds up to arbitrary orders of the form $$ \\ln N+\\gamma+4\\ln 2+\\sum_{s=1}^{2m}"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1309.4564","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-09-18T07:48:17Z","cross_cats_sorted":[],"title_canon_sha256":"be23f984a6cc152eb8686cef5c112715e3a017ab6d38e3c217688fcbf83aa06f","abstract_canon_sha256":"03a7899069dfa4243dfc7267ba281ec891f318b5039043c8ad24b90655e96054"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:30:22.897802Z","signature_b64":"vyze4wIPEnkEo3mo51eySLDqtVfIvFNk/ADp/7nGCH6ILbQMnbEB6MFKgz7lcUzxCqAdf+4IQade7pRLX0QHBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bc8aa5143d2c18f24a28cceb0b3f3b119413278d41862a64096e68f377d21f58","last_reissued_at":"2026-05-18T02:30:22.897191Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:30:22.897191Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotics of Landau constants with optimal error bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Saiyu Liu, Shuaixia Xu, Yuqiu Zhao, Yutian Li","submitted_at":"2013-09-18T07:48:17Z","abstract_excerpt":"We study the asymptotic expansion for the Landau constants $G_n$ $$\\pi G_n\\sim \\ln N + \\gamma+4\\ln 2 + \\sum_{s=1}^\\infty \\frac {\\beta_{2s}}{N^{2s}},~~n\\rightarrow \\infty, $$ where $N=n+3/4$, $\\gamma=0.5772\\cdots$ is Euler's constant, and $(-1)^{s+1}\\beta_{2s}$ are positive rational numbers, given explicitly in an iterative manner. We show that the error due to truncation is bounded in absolute value by, and of the same sign as, the first neglected term for all nonnegative $n$. 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