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:

$A\approx1.2824271291\dots$   (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.

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

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