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:

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

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

${\displaystyle A=\lim _{n\rightarrow \infty }{\frac {H(n)}{n^{{\frac {n^{2}}{2}}+{\frac {n}{2}}+{\frac {1}{12}}}\,e^{-{\frac {n^{2}}{4}}}}}}$

where H(n) = Πⁿ
_k=1 k^k is the hyperfactorial. This formula displays a similarity between A and π which is perhaps best illustrated by noting Stirling's formula:

${\displaystyle {\sqrt {2\pi }}=\lim _{n\to \infty }{\frac {n!}{n^{n+{\frac {1}{2}}}\,e^{-n}}}}$

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 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:

${\displaystyle A=\lim _{n\rightarrow \infty }{\frac {\left(2\pi \right)^{\frac {n}{2}}n^{{\frac {n^{2}}{2}}-{\frac {1}{12}}}e^{-{\frac {3n^{2}}{4}}+{\frac {1}{12}}}}{G(n+1)}}}$.

The Glaisher–Kinkelin constant also appears in evaluations of the derivatives of the Riemann zeta function, such as:

${\displaystyle \zeta '(-1)={\tfrac {1}{12}}-\ln A}$

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

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

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

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

${\displaystyle \prod _{k=1}^{\infty }p_{k}^{\frac {1}{p_{k}^{2}-1}}={\frac {A^{12}}{2\pi e^{\gamma }}},}$

where p_k denotes the kth prime number.

The following are some integrals that involve this constant:

${\displaystyle \int _{0}^{\frac {1}{2}}\ln \Gamma (x)\,dx={\tfrac {3}{2}}\ln A+{\frac {5}{24}}\ln 2+{\tfrac {1}{4}}\ln \pi }$

${\displaystyle \int _{0}^{\infty }{\frac {x\ln x}{e^{2\pi x}-1}}\,dx={\tfrac {1}{2}}\zeta '(-1)={\tfrac {1}{24}}-{\tfrac {1}{2}}\ln A}$

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

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