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:
A ≈ 1.2824271291 … (sequence
A074962 in the OEIS).
The Glaisher–Kinkelin constant A can be given by the limit:
A = lim n → ∞ K ( n + 1 ) n n 2 / 2 + n / 2 + 1 / 12 e − n 2 / 4 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:
2 π = lim n → ∞ n ! e − n n n + 1 2 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:
A = lim n → ∞ ( 2 π ) n / 2 n n 2 / 2 − 1 / 12 e − 3 n 2 / 4 + 1 / 12 G ( n + 1 ) .
The Glaisher–Kinkelin constant also appears in evaluations of the derivatives of the Riemann zeta function, such as:
ζ ′ ( − 1 ) = 1 12 − ln A ∑ k = 2 ∞ ln k k 2 = − ζ ′ ( 2 ) = π 2 6 [ 12 ln A − γ − ln ( 2 π ) ] where γ is the Euler–Mascheroni constant. The latter formula leads directly to the following product found by Glaisher:
∏ k = 1 ∞ k 1 / k 2 = ( A 12 2 π e γ ) π 2 / 6 The following are some integrals that involve this constant:
∫ 0 1 / 2 ln Γ ( x ) d x = 3 2 ln A + 5 24 ln 2 + 1 4 ln π ∫ 0 ∞ x ln x e 2 π x − 1 d x = 1 2 ζ ′ ( − 1 ) = 1 24 − 1 2 ln A A series representation for this constant follows from a series for the Riemann zeta function given by Helmut Hasse.
ln A = 1 8 − 1 2 ∑ n = 0 ∞ 1 n + 1 ∑ k = 0 n ( − 1 ) k ( n k ) ( k + 1 ) 2 ln ( k + 1 ) 