Difference between revisions of "Glaisher–Kinkelin constant"
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[http://mpmath.googlecode.com/svn/data/glaisher.txt The Glaisher–Kinkelin constant to 20,000 decimal places] | [http://mpmath.googlecode.com/svn/data/glaisher.txt The Glaisher–Kinkelin constant to 20,000 decimal places] | ||
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Revision as of 18:59, 24 May 2016
The Glaisher–Kinkelin constant is defined by the formula $$A=\displaystyle\lim_{n \rightarrow \infty} \dfrac{(2\pi)^{\frac{n}{2}}n^{\frac{n^2}{2}-\frac{1}{12}}e^{-\frac{3n^2}{4}+\frac{1}{12}}}{G(n+1)},$$ where $G$ is the Barnes $G$ function.
Properties
Theorem
The following formula holds: $$\zeta'(-1)=\dfrac{1}{12}-\log(A),$$ where $\zeta$ denotes the Riemann zeta function, $A$ denotes the Glaisher–Kinkelin constant, and $\log$ denotes the logarithm.
