Zeta Functions and the Log Behaviour of Combinatorial Sequences

Source abstract

Abstract In this paper, we use the Riemann zeta function ζ (x) and the Bessel zeta function ζ μ (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 {|B 2n |/(2n)!} n ≥ 1 is log-convex, where B n is the n th 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 log θ (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 a n (μ) = 2 2n+1 ( n + 1)!( μ + 1) n ζ μ (2 n ) and conjectured that the sequence {an(μ)} n ≥1 is log-convex for μ = 0 and μ = 1. By proving that ζ μ (x) is log-convex for x > 1 and μ > -1, we show that the sequence {a n ( ≥ )} n >1 is log-convex for any μ > - 1. We introduce another function θ μ ,(x) involving ζ μ (x) and the gamma function Γ(x) and we show that log θ μ (x) is strictly increasing for x > 8e( μ + 2) 2 . This implies that Based on Dobinski’s formula, we prove that 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 and Holder’s inequality in probability theory.