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 can be given by the limit:
where is the K-function. This formula displays a similarity between A and which is perhaps best illustrated by noting Stirling's formula:
which shows that just as is obtained from approximation of the function, A can also be obtained from a similar approximation to the function.
An equivalent definition for A involving the Barnes G-function, given by where is the gamma function is:
The Glaisher–Kinkelin constant also appears in evaluations of the derivatives of the Riemann zeta function, such as:
where is the Euler–Mascheroni constant. The latter formula leads directly to the following product found by Glaisher:
An alternative product formula, defined over the prime numbers, reads
where denotes the th prime number.
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.
