Formula:DLMF:25.11:E24

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r = 1 k - 1 ζ ( s , r k ) = ( k s - 1 ) ζ ( s ) + k s ζ ( s ) ln k superscript subscript r 1 k 1 diffop Hurwitz-zeta 1 s r k superscript k s 1 diffop Riemann-zeta 1 s superscript k s Riemann-zeta s natural-lograrithm k {\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)


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


Note(s)


primes on ζ Hurwitz-zeta {\displaystyle\HurwitzZeta} denote derivatives with respect to s 𝑠 {\displaystyle s}


Proof


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Use
ζ ( s , k a ) = k - s n = 0 k - 1 ζ ( s , a + n k ) Hurwitz-zeta s k a superscript k s superscript subscript n 0 k 1 Hurwitz-zeta s a n k {\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 a = 1 / k 𝑎 1 𝑘 {\displaystyle a=1/k} , multiply by k s superscript 𝑘 𝑠 {\displaystyle k^{s}} and differentiate.



Symbols List


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

Bibliography


Equation (15), Section 25.11 of DLMF.

URL links


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