{"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 $n$. We also show that for given $r\\geq 1$ and sufficiently large $n$, $(-1)^{r-1}\\Delt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.0177","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}