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\approx 1.2824271291\dots

(sequence A074962 in the 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^{1/k^{2}}=\left({\frac {A^{12}}{2\pi e^{\gamma }}}\right)^{\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}(-1)^{k}{\binom {n}{k}}(k+1)^{2}\ln(k+1)

References

Guillera, Jesus; Sondow, Jonathan (2005). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". The Ramanujan Journal. 16 (3): 247–270. arXiv:math.NT/0506319. doi:10.1007/s11139-007-9102-0.
Guillera, Jesus; Sondow, Jonathan (2008). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". Ramanujan Journal. 16 (3): 247&ndash, 270. arXiv:math/0506319. doi:10.1007/s11139-007-9102-0. (Provides a variety of relationships.)
Weisstein, Eric W. "Glaisher–Kinkelin Constant". MathWorld.
Weisstein, Eric W. "Riemann Zeta Function". MathWorld.

External links

The Glaisher–Kinkelin constant to 20,000 decimal places

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