Glaisher–Kinkelin constant

In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted , is a mathematical constant, related to the -function and the Barnes -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:
:$A=\lim_ \frac$
where is the hyperfactorial. This formula displays a similarity between and which is perhaps best illustrated by noting Stirling's formula:
:$\sqrt=\lim_ \frac$
which shows that just as is obtained from approximation of the factorials, can also be obtained from a similar approximation to the hyperfactorials.

An equivalent definition for involving the Barnes -function, given by where is the gamma function is:
:$A=\lim_ \frac$.
The Glaisher–Kinkelin constant also appears in evaluations of the derivatives of the Riemann zeta function, such as:
:$\zeta'(-1)=\tfrac-\ln A$

: $\sum_^\infty \frac=-\zeta'(2)=\frac 6 \left( 12 \ln A - \gamma-\ln 2\pi \right)$

where is the Euler–Mascheroni constant. The latter formula leads directly to the following product found by Glaisher:

: $\prod_^\infty k^\frac = \left(\frac \right)^\frac$

An alternative product formula, defined over the prime numbers, reads

: $\prod_^\infty p_k^\frac = \frac,$

where denotes the th prime number.

The following are some integrals that involve this constant:

: $\int_0^\frac12 \ln\Gamma(x) \, dx = \tfrac 3 2 \ln A+\frac 5 \ln 2+\tfrac 1 4 \ln \pi$

: $\int_0^\infty \frac \, dx = \tfrac 1 2 \zeta'(-1) = \tfrac 1 -\tfrac 1 2 \ln A$

A series representation for this constant follows from a series for the Riemann zeta function given by Helmut Hasse.

: $\ln A=\tfrac 1 8 - \tfrac 1 2 \sum_^\infty \frac 1 \sum_^n (-1)^k \binom n k (k+1)^2 \ln(k+1)$

References

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External links

The Glaisher–Kinkelin constant to 20,000 decimal places

{{DEFAULTSORT:Glaisher-Kinkelin constant
Mathematical constants
Number theory
Glaisher family