Formula:DLMF:25.11:E21

<< Formula:DLMF:25.11:E20

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E22 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{1-2n}{\frac{h}{k}}=\frac{(\psi\left(2n\right)-\ln\left(2\pi k\right))\BernoulliB{2n}@{h/k}}{2n}-\frac{(\psi\left(2n\right)-\ln\left(2\pi\right))\BernoulliB{2n}}{2nk^{2n}}+\frac{(-1)^{n+1}\pi}{(2\pi k)^{2n}}\sum_{r=1}^{k-1}\sin\left(\frac{2\pi rh}{k}\right){\psi^{(2n-1)}}\left(\frac{r}{k}\right)+\frac{(-1)^{n+1}2\cdot(2n-1)!}{(2\pi k)^{2n}}\sum_{r=1}^{k-1}\cos\left(\frac{2\pi rh}{k}\right)\HurwitzZeta^{\prime}@{2n}{\frac{r}{k}}+\frac{\RiemannZeta^{\prime}@{1-2n}}{k^{2n}}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle h,k}}}$ are integers with ${\displaystyle{\displaystyle{\displaystyle 1\leq h\leq k}}}$ &
${\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}}$

Note(s)

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

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\psi}}}$ : psi (or digamma) function : http://dlmf.nist.gov/5.2#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle B_{n}}}}$ : Bernoulli polynomial : http://dlmf.nist.gov/24.2#i
${\displaystyle{\displaystyle{\displaystyle(-1)}}}$ : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1

Bibliography

Equation (21), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E20

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E22 >>