# Formula:DLMF:25.11:E37

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

## Constraint(s)

$\displaystyle {\displaystyle n = 2,3,4,\dots}$ &
$\displaystyle {\displaystyle \realpart{a} \geq 1}$

## Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

## Symbols List

& : logical and
$\displaystyle {\displaystyle \Sigma}$  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
$\displaystyle {\displaystyle (-1)}$  : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
$\displaystyle {\displaystyle \zeta}$  : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
$\displaystyle {\displaystyle \mathrm{ln}}$  : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
$\displaystyle {\displaystyle \Gamma}$  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
$\displaystyle {\displaystyle \Pi}$  : product : http://drmf.wmflabs.org/wiki/Definition:prod
$\displaystyle {\displaystyle \mathrm{e}}$  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
$\displaystyle {\displaystyle \pi}$  : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
$\displaystyle {\displaystyle \mathrm{i}}$  : imaginary unit : http://dlmf.nist.gov/1.9.i
$\displaystyle {\displaystyle \Re {z}}$  : real part : http://dlmf.nist.gov/1.9#E2