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:
The Glaisher–Kinkelin constant A can be given by the limit:
- [math]\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}}} }[/math]
where H(n) = Πnk=1 kk is the hyperfactorial. This formula displays a similarity between A and π which is perhaps best illustrated by noting Stirling's formula:
- [math]\displaystyle{ \sqrt{2\pi}=\lim_{n \to \infty} \frac{n!}{n^{n+\frac12}\,e^{-n}} }[/math]
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−2k=1 k! = [Γ(n)]n−1/K(n) where Γ(n) is the gamma function is:
- [math]\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)} }[/math].
The Glaisher–Kinkelin constant also appears in evaluations of the derivatives of the Riemann zeta function, such as:
- [math]\displaystyle{ \zeta'(-1)=\tfrac{1}{12}-\ln A }[/math]
- [math]\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) }[/math]
where γ is the Euler–Mascheroni constant. The latter formula leads directly to the following product found by Glaisher:
- [math]\displaystyle{ \prod_{k=1}^\infty k^\frac{1}{k^2} = \left(\frac{A^{12}}{2\pi e^\gamma} \right)^\frac{\pi^2}{6} }[/math]
An alternative product formula, defined over the prime numbers, reads [1]
- [math]\displaystyle{ \prod_{k=1}^\infty p_k^\frac{1}{p_k^2-1} = \frac{A^{12}}{2\pi e^\gamma}, }[/math]
where pk denotes the kth prime number.
The following are some integrals that involve this constant:
- [math]\displaystyle{ \int_0^\frac12 \ln\Gamma(x) \, dx = \tfrac 3 2 \ln A+\frac 5 {24} \ln 2+\tfrac 1 4 \ln \pi }[/math]
- [math]\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 }[/math]
A series representation for this constant follows from a series for the Riemann zeta function given by Helmut Hasse.
- [math]\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) }[/math]
References
- ↑ Van Gorder, Robert A. (2012). "Glaisher-Type Products over the Primes". International Journal of Number Theory 08 (2): 543–550. doi:10.1142/S1793042112500297.
- Guillera, Jesus; Sondow, Jonathan (2008). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". The Ramanujan Journal 16 (3): 247–270. 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): 247–270. doi:10.1007/s11139-007-9102-0. (Provides a variety of relationships.)
- Weisstein, Eric W.. "Glaisher–Kinkelin Constant". http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html.
- Weisstein, Eric W.. "Riemann Zeta Function". http://mathworld.wolfram.com/RiemannZetaFunction.html.
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