ZETA FUNCTIONS AND THE LOG BEHAVIOUR OF COMBINATORIAL SEQUENCES

In this paper, we use the Riemann zeta function zeta(x) and the Bessel zeta function zeta(mu)(x) to study the log behaviour of combinatorial sequences. We prove that zeta(x) is log-convex for x > 1. As a consequence, we deduce that the sequence {vertical bar B-2n vertical bar/(2n)!}(n >= 1) is log-convex, where B-n is the nth Bernoulli number. We introduce the function theta(x) = (2 zeta(x)Gamma(x + 1))(1/x), where Gamma(x) is the gamma function, and we show that log theta(x) is strictly increasing for x >= 6. This confirms a conjecture of Sun stating that the sequence {n root vertical bar B2n vertical bar}(n >= 1) is strictly increasing. Amdeberhan et al. 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 >= 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 >= 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 n root a(n)(mu) < (n+1)root a(n+1)(mu) for n > 4e(mu + 2)(2). Based on Dobinski's formula, we prove that n root B-n < (n+1)root Bn+1 for n >= 1, where B-n is the nth Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property of {(n)root B-n}(n >= 1) and Holder's inequality in probability theory.