Zeta Functions and the Log Behaviour of Combinatorial Sequences

In this paper, we use the Riemann zeta function ζ(x) and the Bessel zeta function ζμ(x) to study the log behaviour of combinatorial sequences. We prove that ζ(x) is log-convex for x > 1. As a consequence, we deduce that the sequence {|B2n|/(2n)!}n ≥ 1 is log-convex, where Bn is the nth Bernoulli number. We introduce the function θ(x) = (2ζ(x)Γ(x + 1)) 1/x, where Γ(x) is the gamma function, and we show that logθ(x) is strictly increasing for x ≥ 6. This confirms a conjecture of Sun stating that the sequence is strictly increasing. Amdeberhan et al. defined the numbers an(μ) = 2 2n+1 (n + 1)!(μ+ 1)nζμ(2n) and conjectured that the sequence {an(μ)}n≥1 is log-convex for μ = 0 and μ = 1. By proving that ζμ(x) is log-convex for x > 1 and μ > -1, we show that the sequence {an(≥)}n>1 is log-convex for any μ > - 1. We introduce another function θμ,(x) involving ζμ(x) and the gamma function Γ(x) and we show that logθμ(x) is strictly increasing for x > 8e(μ + 2)2. This implies that Based on Dobinski’s formula, we prove that where Bn is the nth Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property of and Holder’s inequality in probability theory.