# Sums

## Sums

$\displaystyle {\displaystyle \sum_{k \hiderel{=} 2}^\infty \left( \RiemannZeta@{k} - 1 \right) = 1 }$
$\displaystyle {\displaystyle \sum_{k \hiderel{=} 0}^\infty \frac{\EulerGamma@{s+k}}{(k+1)!} \left( \RiemannZeta@{s+k} - 1 \right) = \EulerGamma@{s-1} }$

Constraint(s): $\displaystyle {\displaystyle s \neq 1,0,-1,-2,\dots}$

$\displaystyle {\displaystyle \sum_{k \hiderel{=} 0}^\infty \frac{\EulerGamma@{s+k} \RiemannZeta@{s+k}}{k! \EulerGamma@{s} 2^{s+k}} = (1 - 2^{-s}) \RiemannZeta@{s} }$

Constraint(s): $\displaystyle {\displaystyle s \neq 1}$

$\displaystyle {\displaystyle \sum_{k \hiderel{=} 1}^\infty \frac{\opminus^k}{k} (\RiemannZeta@{nk} - 1) = \ln@{\prod_{j=0}^{n-1} \EulerGamma@{2 - \expe^{(2j+1) \cpi \iunit/n}}} }$

Constraint(s): $\displaystyle {\displaystyle n = 2,3,4,\dots}$

$\displaystyle {\displaystyle \sum_{k \hiderel{=} 2}^\infty \RiemannZeta@{k} z^k = - \EulerConstant z - z \digamma@{1-z} }$

Constraint(s): $\displaystyle {\displaystyle |z| < 1}$

$\displaystyle {\displaystyle \sum_{k \hiderel{=} 0}^\infty \RiemannZeta@{2k} z^{2k} = - \tfrac{1}{2} \cpi z \cot@{\cpi z} }$

Constraint(s): $\displaystyle {\displaystyle |z| < 1}$

$\displaystyle {\displaystyle \sum_{k \hiderel{=} 2}^\infty \frac{\RiemannZeta@{k}}{k} z^k = -\EulerConstant z + \ln@@{\EulerGamma@{1-z}} }$

Constraint(s): $\displaystyle {\displaystyle |z| < 1}$

$\displaystyle {\displaystyle \sum_{k \hiderel{=} 1}^\infty \frac{\RiemannZeta@{2k}}{k} z^{2k} = \ln@{\frac{\cpi z}{\sin@{\cpi z}}} }$

Constraint(s): $\displaystyle {\displaystyle |z| < 1}$

$\displaystyle {\displaystyle \sum_{k \hiderel{=} 1}^\infty \frac{\RiemannZeta@{2k}}{(2k+1)2^{2k}} = \frac{1}{2} - \frac{1}{2} \ln 2 }$
$\displaystyle {\displaystyle \sum_{k \hiderel{=} 1}^\infty \frac{\RiemannZeta@{2k}}{(2k+1) (2k+2) 2^{2k}} = \frac{1}{4} - \frac{7}{4 \cpi^2} \RiemannZeta@{3} }$