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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 $x\\geq 6$. 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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 $x\\geq 6$. 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