# Glaisher–Kinkelin constant

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In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted A, is a mathematical constant, related to the K-function and the Barnes G-function. The constant appears in a number of sums and integrals, especially those involving Gamma functions and zeta functions. It is named after mathematicians James Whitbread Lee Glaisher and Hermann Kinkelin.

Its approximate value is:

${\displaystyle A\approx 1.2824271291\dots }$   (sequence A074962 in OEIS).

The Glaisher–Kinkelin constant ${\displaystyle A}$ can be given by the limit:

${\displaystyle A=\lim _{n\rightarrow \infty }{\frac {K(n+1)}{n^{n^{2}/2+n/2+1/12}e^{-n^{2}/4}}}}$
${\displaystyle A=\lim _{n\rightarrow \infty }{\frac {(2\pi )^{n/2}n^{n^{2}/2-1/12}e^{-3n^{2}/4+1/12}}{G(n+1)}}}$.

The Glaisher–Kinkelin constant also appears in the Riemann zeta function, such as:

${\displaystyle \zeta ^{\prime }(-1)={\frac {1}{12}}-\ln A}$
${\displaystyle \sum _{k=2}^{\infty }{\frac {\ln k}{k^{2}}}=-\zeta ^{\prime }(2)={\frac {\pi ^{2}}{6}}\left[12\ln A-\gamma -\ln(2\pi )\right]}$

where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant.
Some integrals involve this constant:

${\displaystyle \int _{0}^{1/2}\ln \Gamma (x)dx={\frac {3}{2}}\ln A+{\frac {5}{24}}\ln 2+{\frac {1}{4}}\ln \pi }$
${\displaystyle \int _{0}^{\infty }{\frac {x\ln x}{e^{2\pi x}-1}}dx={\frac {1}{2}}\zeta ^{\prime }(-1)={\frac {1}{24}}-{\frac {1}{2}}\ln A}$

A series representation for this constant follows from a series for the Riemann zeta function given by Helmut Hasse.

${\displaystyle \ln A={\frac {1}{8}}-{\frac {1}{2}}\sum _{n=0}^{\infty }{\frac {1}{n+1}}\sum _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}\left(k+1\right)^{2}\ln(k+1)}$

## References

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