{"paper":{"title":"Zeta Functions and the Log-behavior of Combinatorial Sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Jeremy J.F. Guo, Larry X.W. Wang, William Y.C. Chen","submitted_at":"2012-08-26T10:37:36Z","abstract_excerpt":"In this paper, we use the Riemann zeta function $\\zeta(x)$ and the Bessel zeta function $\\zeta_{\\mu}(x)$ to study the log-behavior of combinatorial sequences. We prove that $\\zeta(x)$ is log-convex for $x>1$. As a consequence, we deduce that the sequence $\\{|B_{2n}|/(2n)!\\}_{n\\geq 1}$ is log-convex, where $B_n$ is the $n$-th Bernoulli number. We introduce the function $\\theta(x)=(2\\zeta(x)\\Gamma(x+1))^{\\frac{1}{x}}$, where $\\Gamma(x)$ is the gamma function, and we show that $\\log \\theta(x)$ is strictly increasing for $x\\geq 6$. This confirms a conjecture of Sun stating that the sequence $\\{\\sq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.5213","kind":"arxiv","version":3},"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"}