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 H(n) = Πn
k=1 kk is the hyperfactorial. 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 factorials, A can also be obtained from a similar approximation to the hyperfactorials.
An equivalent definition for A involving the Barnes G-function, given by G(n) = Πn−2
k=1 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:
An alternative product formula, defined over the prime numbers, reads 
where pk denotes the kth 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.
- ^ Van Gorder, Robert A. (2012). "Glaisher-Type Products over the Primes". International Journal of Number Theory. 08 (2): 543–550. doi:10.1142/S1793042112500297.
- Guillera, Jesus; Sondow, Jonathan (2008). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". The Ramanujan Journal. 16 (3): 247–270. arXiv:math.NT/0506319. doi:10.1007/s11139-007-9102-0. S2CID 14910435.
- 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. arXiv:math/0506319. doi:10.1007/s11139-007-9102-0. S2CID 14910435. (Provides a variety of relationships.)
- Weisstein, Eric W. "Glaisher–Kinkelin Constant". MathWorld.
- Weisstein, Eric W. "Riemann Zeta Function". MathWorld.