Hurwitz zeta
The Hurwitz zeta function is a generalization of the Riemann zeta function defined initially for $\mathrm{Re}(s)>1$ and $\mathrm{Re}(a)>0$ by $$\zeta(s,a)= \displaystyle\sum_{n=0}^{\infty} \dfrac{1}{(n+a)^s}.$$
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Properties
Theorem: The function $\zeta(s,a)$ is absolutely convergent for all $s$ with $\mathrm{Re}(s)>1$ and $a$ with $\mathrm{Re}(a)>0$..
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Theorem: The function $\zeta(s,a)$ is analytic for all $s$ except for a simple pole at $s=1$ with residue $1$.
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Theorem
The following formula holds: $$\Gamma(s)\zeta(s,a) = \displaystyle\int_0^{\infty} \dfrac{x^{s-1}e^{-ax}}{1-e^{-x}} \mathrm{d}x,$$ where $\Gamma$ denotes the gamma function and $\zeta$ denotes the Hurwitz zeta function.
Proof
References
Theorem
The following formula holds: $$B_n(x)=-n \zeta(1-n,x),$$ where $B_n$ denotes the Bernoulli polynomial and $\zeta$ denotes the Hurwitz zeta function.
Proof
References
Theorem
The following formula holds: $$K=\dfrac{\pi}{24} -\dfrac{\pi}{2}\log(A)+4\pi \zeta' \left(-1 , \dfrac{1}{4} \right),$$ where $K$ is Catalan's constant, $A$ is the Glaisher–Kinkelin constant, and $\zeta'$ denotes the partial derivative of the Hurwitz zeta function with respect to the first argument.
