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 A can be given by the limit:

where K ( n ) = ∏ 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:

which shows that just as π is obtained from approximation of the function ∏ k = 1 n k , A can also be obtained from a similar approximation to the function ∏ k = 1 n k k .
An equivalent definition for A involving the Barnes G-function, given by G ( n ) = ∏ k = 1 n − 2 k ! = [ Γ ( n ) ] n − 1 K ( n ) where Γ ( n ) is the gamma function is:

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

where γ is the Euler–Mascheroni constant. The latter formula leads directly to the following product found by Glaisher:

The following are some integrals that involve this constant:

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