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:

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

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.

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

References

• Guillera, Jesus; Sondow, Jonathan (2005). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". arXiv:math.NT/0506319.
• 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): 247–270. 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