Glaisher–Kinkelin constant

The Glaisher–Kinkelin constant, usually denoted by the symbol \(A\), is a mathematical constant which is approximately equal to

\[ 1.2824271291006226368753425688697917277676889273250011920637400217. \]

It can be expressed as the limit, \(A = \displaystyle \lim_{n\to\infty} \dfrac{1^1 \times 2^2 \times 3^3 \times \cdots \times n^n}{\exp_n \left( \dfrac {n^2}2 + \dfrac m2 + \dfrac1{12} \right) e^{-n^2/4}} \), where \(\exp_P Q = P^Q \) and \(e\) denotes Euler's number.

The Glaisher-Kinkelin constant can also be evaulated as the derivative of the Riemann zeta function, \(A = \exp \left [-\dfrac{\zeta'(2)}{2\pi^2} + \dfrac{\ln(2\pi)}{12} + \dfrac \gamma 2 \right ] \), where \( \gamma\) denotes the Euler-Mascheroni constant.

It can also be expressed as \( \displaystyle A = 2^{7/36} \pi^{-1/6} \exp \left [\dfrac13 + \dfrac23 \int_0^{1/2} \ln( \Gamma(x+1)) \, dx \right ] \).