Zeta Functions and the Log-behavior of Combinatorial Sequences
William Y. C. Chen, Jeremy J. F. Guo and Larry X. W. Wang
Abstract: 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
logθ(x) is strictly increasing for x≥6.
This confirms a conjecture of Sun stating that the sequence
AMS Classification: 05A20, 11B68 Keywords: log-convexity, Riemann zeta function, Bernoulli number, Bell number, Bessel zeta function, Narayana number Download: PDF |