# Glaisher–Kinkelin constant

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:

$A\approx1.2824271291\dots$   (sequence A074962 in OEIS).

The Glaisher–Kinkelin constant $A$ can be given by the limit:

$A=\lim_{n\rightarrow\infty} \frac{K(n+1)}{n^{n^2/2+n/2+1/12} e^{-n^2/4}}$

where $K(n)=\prod_{k=1}^{n-1} k^k$ is the K-function. An equivalent form involving the Barnes G-function, given by $G(n)=\prod_{k=1}^{n-2}k!=\frac{\left[\Gamma(n)\right]^{n-1}}{K(n)}$ where $\Gamma(n)$ is the gamma function is:

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

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

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

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

$\int_0^{1/2} \ln\Gamma(x)dx=\frac{3}{2} \ln A+\frac{5}{24} \ln 2+\frac{1}{4} \ln \pi$
$\int_0^\infty \frac{x \ln x}{e^{2 \pi x}-1}dx=\frac{1}{2} \zeta^{\prime}(-1)=\frac{1}{24}-\frac{1}{2}\ln A$

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

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

## References

• Guillera, Jesus; Sondow, Jonathan (2005). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". arXiv:math.NT/0506319.
• Guillera, Jesus; Sondow, Jonathan (2008). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". Ramanujan Journal 16 (3): 247–270. doi:10.1007/s11139-007-9102-0. (Provides a variety of relationships.)