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Monotonicity Properties of the Hurwitz Zeta Function


Published:20050901
Printed: Sep 2005
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Abstract
Let
$$
\zeta(s,x)=\sum_{n=0}^{\infty}\frac{1}{(n+x)^s} \quad{(s>1,\, x>0)}
$$
be the Hurwitz zeta function and let
$$
Q(x)=Q(x;\alpha,\beta;a,b)=\frac{(\zeta(\alpha,x))^a}{(\zeta(\beta,x))^b},
$$
where $\alpha, \beta>1$
and $a,b>0$ are real numbers. We prove:
(i) The function $Q$ is decreasing on $(0,\infty)$ if{}f $\alpha a\beta b\geq \max(ab,0)$.
(ii) $Q$ is increasing on $(0,\infty)$ if{}f $\alpha a\beta b\leq
\min(ab,0)$.
An application of part (i) reveals that for all $x>0$ the function $s\mapsto [(s1)\zeta(s,x)]^{1/(s1)}$ is decreasing on $(1,\infty)$. This settles
a conjecture of Bastien and Rogalski.