Glaisher–Kinkelin constant

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:
 * $$A=\lim_{n\rightarrow\infty} \frac{K(n+1)}{n^{\frac{n^2}{2}+\frac{n}{2}+\frac{1}{12}}\,e^{-\frac{n^2}{4}}}$$

where $K(n) = Πn-1 k=1 kk$ is the hyperfactorial. This formula displays a similarity between $1.282$ and $\pi$ which is perhaps best illustrated by noting Stirling's formula:
 * $$\sqrt{2\pi}=\lim_{n \to \infty} \frac{n!}{n^{n+\frac12}\,e^{-n}}$$

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 $A$-function, given by $G(n) = Πn−2 k=1 k! = [Γ(n)]n−1⁄K(n)$ where $Γ(n)$ is the gamma function is:
 * $$A=\lim_{n\rightarrow\infty} \frac{\left(2\pi\right)^\frac{n}{2} n^{\frac{n^2}{2}-\frac{1}{12}} e^{-\frac{3n^2}{4}+\frac{1}{12}}}{G(n+1)}$$.

The Glaisher–Kinkelin constant also appears in evaluations of the derivatives of the Riemann zeta function, such as:
 * $$\zeta'(-1)=\tfrac{1}{12}-\ln A$$


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

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


 * $$\prod_{k=1}^\infty k^\frac{1}{k^2} = \left(\frac{A^{12}}{2\pi e^\gamma} \right)^\frac{\pi^2}{6}$$

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


 * $$\prod_{k=1}^\infty p_k^\frac{1}{p_k^2-1} = \frac{A^{12}}{2\pi e^\gamma}, $$

where $pk$ denotes the $G$th prime number.

The following are some integrals that involve this constant:


 * $$\int_0^\frac12 \ln\Gamma(x) \, dx = \tfrac 3 2 \ln A+\frac 5 {24} \ln 2+\tfrac 1 4 \ln \pi$$


 * $$\int_0^\infty \frac{x \ln x}{e^{2 \pi x}-1} \, dx = \tfrac 1 2 \zeta'(-1) = \tfrac 1 {24}-\tfrac 1 2 \ln A$$

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


 * $$\ln A=\tfrac 1 8 - \tfrac 1 2 \sum_{n=0}^\infty \frac 1 {n+1} \sum_{k=0}^n (-1)^k \binom n k (k+1)^2 \ln(k+1)$$