Formula:DLMF:25.8:E4

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k = 1 ( - 1 ) k k ( \RiemannZeta @ n k - 1 ) = ln ( j = 0 n - 1 Γ ( 2 - e ( 2 j + 1 ) π i / n ) ) superscript subscript 𝑘 1 superscript 1 𝑘 𝑘 \RiemannZeta @ 𝑛 𝑘 1 superscript subscript product 𝑗 0 𝑛 1 Euler-Gamma 2 2 𝑗 1 imaginary-unit 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k% }(\RiemannZeta@{nk}-1)=\ln\left(\prod_{j=0}^{n-1}\Gamma\left(2-{\mathrm{e}^{(2% j+1)\pi\mathrm{i}/n}}\right)\right)}}}

Constraint(s)

n = 2 , 3 , 4 , 𝑛 2 3 4 {\displaystyle{\displaystyle{\displaystyle n=2,3,4,\dots}}}


Proof

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Symbols List

Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( - 1 ) 1 {\displaystyle{\displaystyle{\displaystyle(-1)}}}  : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
ζ 𝜁 {\displaystyle{\displaystyle{\displaystyle\zeta}}}  : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
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}}}  : product : http://drmf.wmflabs.org/wiki/Definition:prod
Γ Γ {\displaystyle{\displaystyle{\displaystyle\Gamma}}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
e e {\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
π 𝜋 {\displaystyle{\displaystyle{\displaystyle\pi}}}  : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation (4), Section 25.8 of DLMF.

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