# Formula:DLMF:25.11:E24

${\displaystyle\sum_{r=1}^{k-1}{\mathop{\zeta\/}\nolimits^{\prime}}\!\left(s,% \frac{r}{k}\right)=(k^{s}-1){\mathop{\zeta\/}\nolimits^{\prime}}\!\left(s% \right)+k^{s}\mathop{\zeta\/}\nolimits\!\left(s\right)\mathop{\ln\/}\nolimits k}$

## Constraint(s)

${\displaystyle s\neq 1}$ & ${\displaystyle k=1,2,3,\dots}$

## Note(s)

primes on ${\displaystyle\HurwitzZeta}$ denote derivatives with respect to ${\displaystyle s}$

## Proof

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

Use
${\displaystyle\mathop{\zeta\/}\nolimits\!\left(s,ka\right)=k^{-s}\*\sum_{n=0}^% {k-1}\mathop{\zeta\/}\nolimits\!\left(s,a+\frac{n}{k}\right)}$
with ${\displaystyle a=1/k}$, multiply by ${\displaystyle k^{s}}$ and differentiate.

## Symbols List

${\displaystyle\zeta}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle\zeta}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle\mathrm{ln}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2