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Search: MSC category 11M35 ( Hurwitz and Lerch zeta functions )

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1. CMB 2007 (vol 50 pp. 11)

Borwein, David; Borwein, Jonathan
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)

Alzer, Horst
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

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