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:

A = 1.28242712910062263687...   (sequence A074962 in the OEIS).

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

where K(n) = Πn−1
k=1
kk
is the K-function. 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 function Πn
k=1
k
, A can also be obtained from a similar approximation to the function Πn
k=1
kk
.

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.

References[edit]

  1. ^ Van Gorder, Robert A. (2012). "Glaisher-Type Products over the Primes". International Journal of Number Theory. 08 (2): 543–550. doi:10.1142/S1793042112500297.

External links[edit]