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) = Î n-1
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 [1]
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.
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) = Î n-1
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 [1]
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.