Zeta Functions and the Log-behavior of Combinatorial Sequences

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In this paper, we use the Riemann zeta function ζ(x) and the Bessel zeta function ζμ(x) to study the log-behavior 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 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 θ(x) is strictly increasing for x≥ 6. This confirms a conjecture of Sun stating that the sequence {√[n] |B₂ₙ|}ₙ≥ ₁ is strictly increasing. Amdeberhan, Moll and Vignat defined the numbers aₙ(μ)=2²ⁿ⁺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(μ)}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 θμ(x) is strictly increasing for x>8e(μ+2)². This implies that √[n]aₙ(μ)<√[n+1]aₙ₊₁(μ) for n> 4e(μ+2)². Based on Dobinski's formula, we prove that √[n]Bₙ<√[n+1]Bₙ₊₁ for n≥ 1, where Bₙ is the n-th Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property of {√[n]Bₙ}ₙ≥ ₁ and Hölder's inequality in probability theory.