Formula:DLMF:25.11:E21

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\HurwitzZeta ′ ⁢ @ ⁢ 1 - 2 ⁢ n ⁢ h k = ( ψ ⁡ ( 2 ⁢ n ) - ln ⁡ ( 2 ⁢ π ⁢ k ) ) ⁢ \BernoulliB ⁢ 2 ⁢ n ⁢ @ ⁢ h / k 2 ⁢ n - ( ψ ⁡ ( 2 ⁢ n ) - ln ⁡ ( 2 ⁢ π ) ) ⁢ \BernoulliB ⁢ 2 ⁢ n 2 ⁢ n ⁢ k 2 ⁢ n + ( - 1 ) n + 1 ⁢ π ( 2 ⁢ π ⁢ k ) 2 ⁢ n ⁢ ∑ r = 1 k - 1 sin ⁡ ( 2 ⁢ π ⁢ r ⁢ h k ) ⁢ ψ ( 2 ⁢ n - 1 ) ⁡ ( r k ) + ( - 1 ) n + 1 ⁢ 2 ⋅ ( 2 ⁢ n - 1 ) ! ( 2 ⁢ π ⁢ k ) 2 ⁢ n ⁢ ∑ r = 1 k - 1 cos ⁡ ( 2 ⁢ π ⁢ r ⁢ h k ) ⁢ \HurwitzZeta ′ ⁢ @ ⁢ 2 ⁢ n ⁢ r k + \RiemannZeta ′ ⁢ @ ⁢ 1 - 2 ⁢ n k 2 ⁢ n superscript \HurwitzZeta ′ @ 1 2 𝑛 ℎ 𝑘 digamma 2 𝑛 2 𝑘 \BernoulliB 2 𝑛 @ ℎ 𝑘 2 𝑛 digamma 2 𝑛 2 \BernoulliB 2 𝑛 2 𝑛 superscript 𝑘 2 𝑛 superscript 1 𝑛 1 superscript 2 𝑘 2 𝑛 superscript subscript 𝑟 1 𝑘 1 2 𝑟 ℎ 𝑘 digamma 2 𝑛 1 𝑟 𝑘 ⋅ superscript 1 𝑛 1 2 2 𝑛 1 superscript 2 𝑘 2 𝑛 superscript subscript 𝑟 1 𝑘 1 2 𝑟 ℎ 𝑘 superscript \HurwitzZeta ′ @ 2 𝑛 𝑟 𝑘 superscript \RiemannZeta ′ @ 1 2 𝑛 superscript 𝑘 2 𝑛 {\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)

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


Note(s)

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

Symbols List

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