In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted , is a mathematical constant, related to special functions like the -function and the Barnes -function. The constant also appears in a number of sums and integrals, especially those involving the gamma function and the Riemann zeta function. It is named after mathematicians James Whitbread Lee Glaisher and Hermann Kinkelin.
Its approximate value is:
: = ... .
Glaisher's constant plays a role both in mathematics and in physics. It appears when giving a closed form expression for Porter's constant, when estimating the efficiency of the Euclidean algorithm. It also is connected to solutions of Painlevé differential equations and the Gaudin model.
:<math>A=\lim_{n\rightarrow\infty} \frac{H(n)}{n^{\tfrac{n^2}{2}+\tfrac{n}{2}+\tfrac{1}{12\,e^{-\tfrac{n^2}{4}</math>
where <math>H(n)</math> is the hyperfactorial:<math display="block">
H(n)= \prod_{i=1}^{n} i^i = 1^1\cdot 2^2\cdot 3^3 \cdot {...}\cdot n^n</math>An analogous limit, presenting a similarity between <math>A</math> and <math>\sqrt{2\pi}</math>, is given by Stirling's formula as:
:<math>\sqrt{2\pi}=\lim_{n \to \infty} \frac{n!}{n^{n+\frac12}\,e^{-n</math>
with<math display="block">
n!= \prod_{i=1}^{n} i = 1 \cdot 2\cdot 3\cdot {...} \cdot n</math>which shows that just as π is obtained from approximation of the factorials, A is obtained from the approximation of the hyperfactorials.
Relation to special functions
Just as the factorials can be extended to the complex numbers by the gamma function such that <math>\Gamma(n)=(n-1)!</math> for positive integers n, the hyperfactorials can be extended by the K-function with <math>K(n)= H(n-1)</math> also for positive integers n, where:
:<math>K(z)=(2\pi)^{-\frac{z-1}2} \exp\left[\binom{z}{2}+\int_0^{z-1} \ln \Gamma(t + 1)\,dt\right]</math>
This gives:
:<math>A=\lim_{n\rightarrow\infty} \frac{K(n+1)}{n^{\tfrac{n^2}{2}+\tfrac{n}{2}+\tfrac{1}{12\,e^{-\tfrac{n^2}{4}</math>.
A related function is the Barnes -function which is given by
:<math>G(n)=\frac{(\Gamma(n))^{n-1{K(n)}</math>
and for which a similar limit exists:
:<math>K(1/2) = \frac{A^{3/2{2^{1/24}e^{1/8</math>
:<math>K(1/4) = A^{9/8}\exp\left(\frac{G}{4\pi}-\frac{3}{32}\right)</math>
:<math>G(1/2) = \frac{2^{1/24}e^{1/8{A^{3/2}\pi^{1/4</math>
:<math>G(1/4) = \frac{1}{2^{9/16}A^{9/8}\pi^{3/16}\varpi^{3/8\exp\left(\frac{3}{32}-\frac{G}{4\pi}\right)</math>
with <math>G</math> being Catalan's constant and <math>\varpi=\frac{\Gamma(1/4)^2}{2\sqrt{2\pi</math> being the lemniscate constant.
Similar to the gamma function, there exists a multiplication formula for the K-Function. It involves Glaisher's constant:
: <math>\ln G(z+1) = \frac{z^2}{2} \ln z - \frac{3z^2}{4} + \frac{z}{2}\ln 2\pi -\frac{1}{12} \ln z + \left(\frac{1}{12}-\ln A \right)+\sum_{k=1}^N \frac{B_{2k + 2{4k\left(k + 1\right)z^{2k+O\left(\frac{1}{z^{2N + 2\right)</math>
The Glaisher-Kinkelin constant is related to the derivatives of the Euler-constant function:
: <math>\gamma'(-1)= \frac{11}{6}\ln2 + 6\ln A - \frac32 \ln\pi - 1</math>
: <math>\gamma(-1) = \frac{10}{3}\ln2 + 24\ln A - 4 \ln\pi - \frac{7\zeta(3)}{2\pi^2}- \frac{13}{4}</math>
<math>A</math> also is related to the Lerch transcendent:
:<math>\frac{\partial\Phi}{\partial s}(-1,-1,1)=3\ln A - \frac13\ln2 - \frac 14</math>
Glaisher's constant may be used to give values of the derivative of the Riemann zeta function as closed form expressions, such as:
:<math>\zeta'(-1)=\frac{1}{12}-\ln A</math>
:<math>\zeta'(2)=\frac{\pi^2} 6 \left( \gamma+\ln 2\pi - 12 \ln A \right)</math>
where is the Euler–Mascheroni constant.
Series expressions
The above formula for <math>\zeta'(2)</math> gives the following series:
:<math>\prod_{p \text{ prime p^\frac{1}{p^2-1} = \frac{A^{12{2\pi e^\gamma}, </math>
Another product is given by:
:<math>\sum_{k=1}^\infty \frac{\text{Ci}(2k\pi)}{k^2}=\frac{\pi^2}{2}(4\ln A -1)</math>
Helmut Hasse gave another series representation for the logarithm of Glaisher's constant, following from a series for the Riemann zeta function:
:<math>\ln A=\frac 1 8 - \frac 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)</math>
Integrals
The following are some definite integrals involving Glaisher's constant:
:<math> \int_0^z \ln \Gamma(x)\,dx=\frac{z(1-z)}{2}+\frac{z}{2}\ln 2\pi +z\ln\Gamma(z) -\ln G(1+z) </math>
We further have:<math display="block">
\int_0^\infty \frac{(1-e^{-x/2})(x \coth \tfrac x2 -2)}{x^3} dx= 3\ln A - \frac 13 \ln2 - \frac 18</math>and<math display="block">
\int_0^\infty \frac{(8-3x)e^x-8e^{x/2}-x}{4x^2e^{x}(e^x-1)} dx = 3\ln A - \frac{7}{12}\ln2 + \frac 12 \ln \pi -1</math>A double integral is given by:
:<math> A_k=\exp\left(\frac{B_{k+1{k+1}H_k-\zeta'(-k)\right) </math>
with the harmonic numbers <math>H_k</math> and <math>H_0=0</math>.
Because of the formula
:<math> \zeta'(-2m)=(-1)^m \frac{(2m)!}{2(2\pi)^{2m\zeta(2m+1) </math>
for <math>m > 0</math>, there exist closed form expressions for <math>A_{k}</math> with even <math>k=2m</math> in terms of the values of the Riemann zeta function such as: