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:

  (sequence A074962 in OEIS).

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

where is the K-function. An equivalent form involving the Barnes G-function, given by where is the gamma function is:


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

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

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


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