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:

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

where ${\displaystyle K(n)=\prod _{k=1}^{n-1}k^{k}}$ is the K-function. This formula displays a similarity between A and π which is perhaps best illustrated by noting Stirling's formula:

...
Wikipedia