Zeta Functions and the Log-behavior of Combinatorial Sequences

Abstract

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 $\{\sqrt[n] {|B_{2n}}|\}_{n\geq 1}$ is strictly increasing. Amdeberhan, Moll and Vignat defined the numbers $a_n(\mu)=2^{2n+1}(n+1)!(\mu+1)_n\zeta_{\mu}(2n)$ and conjectured that the sequence $\{a_n(\mu)\}_{n\geq 1}$ is log-convex for $\mu=0$ and $\mu=1$. By proving that $\zeta_{\mu}(x)$ is log-convex for $x>1$ and $\mu>-1$, we show that the sequence $\{a_n(\mu)\}_{n\geq 1}$ is log-convex for any $\mu>-1$. We introduce another function $\theta_{\mu}(x)$ involving $\zeta_{\mu}(x)$ and the gamma function $\Gamma(x)$ and we show that $\log \theta_{\mu}(x)$ is strictly increasing for $x>8e(\mu+2)^2$. This implies that $\sqrt[n]{a_n(\mu)}<\sqrt[n+1]{a_{n+1}(\mu)}$ for $n> 4e(\mu+2)^2$. Based on Dobinski's formula, we prove that $\sqrt[n]{B_n}<\sqrt[n+1]{B_{n+1}}$ for $n\geq 1$, where $B_n$ is the $n$-th Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property of $\{\sqrt[n]{B_n}\}_{n\geq 1}$ and H\"{o}lder's inequality in probability theory.