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Moreover, they conjectured that a sharper inequality $\\frac{p(n-1)}{p(n)}\\left( 1+\\frac{\\pi}{\\sqrt{24}n^{3/2}}\\right) > \\frac{p(n)}{p(n+1)}$ holds for $n\\geq 45$. In this paper, we prove the conjecture of Desalvo and Pak by giving an upper bound for $-\\Delta^{2} \\log p(n-1)$, where $\\Delta$ is the difference operator with respect to $n$. We also show that for given $r\\geq 1$ and sufficiently large $n$, $(-1)^{r-1}\\Delt"},"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":"1407.0177","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-01T10:37:52Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d3d58a6da2e65f427325b3438abc3744e51c0da652b55aaf68da767bb4891d94","abstract_canon_sha256":"748458812cebcac90e10fb766b9fb437300dc85d08bfc139bd752cd67daca465"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:34.954126Z","signature_b64":"Fcxuu9bThiWWP6JHGGtz7QjYjj9NbiTqUxxE/eG2cGNs1u5s5i6dtjLHnFwk1/2TpB+naFMZTEiEnfAdnMzqAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0dd1b79674c9bcb0c4a29ef4dae9004ee5d4d3d47b20c3b0100193548fc93e3e","last_reissued_at":"2026-05-18T02:48:34.953299Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:34.953299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Finite Differences of the Logarithm of the Partition Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Gary Y.B. Xie, Larry X.W. Wang, William Y.C. Chen","submitted_at":"2014-07-01T10:37:52Z","abstract_excerpt":"Let $p(n)$ denote the partition function. DeSalvo and Pak proved that $\\frac{p(n-1)}{p(n)}\\left(1+\\frac{1}{n}\\right)> \\frac{p(n)}{p(n+1)}$ for $n\\geq 2$, as conjectured by Chen. Moreover, they conjectured that a sharper inequality $\\frac{p(n-1)}{p(n)}\\left( 1+\\frac{\\pi}{\\sqrt{24}n^{3/2}}\\right) > \\frac{p(n)}{p(n+1)}$ holds for $n\\geq 45$. In this paper, we prove the conjecture of Desalvo and Pak by giving an upper bound for $-\\Delta^{2} \\log p(n-1)$, where $\\Delta$ is the difference operator with respect to $n$. 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