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1. CMB 2007 (vol 50 pp. 11)
van der Pol Expansions of L-Series We provide concise series representations for various
L-series integrals. Different techniques are needed below and above
the abscissa of absolute convergence of the underlying L-series.
Keywords:Dirichlet series integrals, Hurwitz zeta functions, Plancherel theorems, L-series Categories:11M35, 11M41, 30B50 |
2. CMB 2005 (vol 48 pp. 333)
Monotonicity Properties of the Hurwitz Zeta Function 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 $\alpha a-\beta b\geq \max(a-b,0)$.
(ii) $Q$ is increasing on $(0,\infty)$ if{}f $\alpha a-\beta b\leq
\min(a-b,0)$.
An application of part (i) reveals that for all $x>0$ the function $s\mapsto [(s-1)\zeta(s,x)]^{1/(s-1)}$ is decreasing on $(1,\infty)$. This settles
a conjecture of Bastien and Rogalski.
Categories:11M35, 26D15 |