Glaisher–Kinkelin constant

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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\approx1.2824271291\dots   (sequence A074962 in OEIS).

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

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

where K(n)=\prod_{k=1}^{n-1} k^k is the K-function. This formula displays a similarity between A and π which is perhaps best illustrated by noting Stirling's formula:

\sqrt{2\pi}=\lim_{n \to \infty} \frac{n!}{e^{-n}n^{n+\frac{1}{2}}}

which shows that just as π is obtained from approximation of the function \prod_{k=1}^{n} k, A can also be obtained from a similar approximation to the function \prod_{k=1}^{n} k^k.
An equivalent definition for A involving the Barnes G-function, given by G(n)=\prod_{k=1}^{n-2}k!=\frac{\left[\Gamma(n)\right]^{n-1}}{K(n)} where \Gamma(n) is the gamma function is:

A=\lim_{n\rightarrow\infty} \frac{(2\pi)^{n/2} n^{n^2/2-1/12} e^{-3n^2/4+1/12}}{G(n+1)}.

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

\zeta^{\prime}(-1)=\frac{1}{12}-\ln A
\sum_{k=2}^\infty \frac{\ln k}{k^2}=-\zeta^{\prime}(2)=\frac{\pi^2}{6}\left[12\ln A-\gamma-\ln(2\pi)\right]

where \gamma 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}}

The following are some integrals that involve this constant:

\int_0^{1/2} \ln\Gamma(x)dx=\frac{3}{2} \ln A+\frac{5}{24} \ln 2+\frac{1}{4} \ln \pi
\int_0^\infty \frac{x \ln x}{e^{2 \pi x}-1}dx=\frac{1}{2} \zeta^{\prime}(-1)=\frac{1}{24}-\frac{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=\frac{1}{8}-\frac{1}{2} \sum_{n=0}^\infty \frac{1}{n+1} \sum_{k=0}^n \left(-1\right)^k \binom{n}{k} \left(k+1\right)^2 \ln(k+1)

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