Glaisher-Kinkelin's constant

The Glaisher-Kinkelin constant , often just Glaisher's constant , is a mathematical constant that occurs in some sums and integrals , especially in connection with the gamma function and the Riemann zeta function . It is named after JWL Glaisher and Hermann Kinkelin .

Approximate value

The Glaisher-Kinkelin constant is usually denoted by. Is an approximation ${\displaystyle A}$

${\displaystyle A=1,28242{\text{ }}71291{\text{ }}00622{\text{ }}63687{\text{ }}53425{\text{ }}68869{\text{ }}79172{\text{ }}77676{\text{ }}88927{\text{ }}32500{\text{ }}11920{\text{ }}63740{\text{ }}02174{\text{ }}04063{\text{ }}08858{\text{ }}82646{\text{ }}11297{\text{ }}36491{\text{ }}95820{\text{ }}23743{\text{ }}...}$

The individual decimal places form the sequence A074962 in OEIS .

Definitions

One possible definition of is ${\displaystyle A}$

${\displaystyle A=\lim _{n\to \infty }{\frac {K(n+1)}{n^{{\frac {1}{2}}n^{2}+{\frac {1}{2}}n+{\frac {1}{12}}}\,e^{-{\frac {1}{4}}n^{2}}}}}$

with the K function ${\displaystyle K(n+1)=1^{1}\cdot 2^{2}\cdot 3^{3}\cdots n^{n}.}$

Another definition is

${\displaystyle A=\exp \left({\tfrac {1}{12}}-\zeta ^{\prime }(-1)\right),}$

which is related to the derivation of the Riemann zeta function ${\displaystyle \zeta }$ .

Another definition using the circle number is: ${\displaystyle \pi }$

${\displaystyle A=\lim _{n\to \infty }{\frac {n^{{\frac {1}{2}}n^{2}-{\frac {1}{12}}}\,(2\pi )^{{\frac {1}{2}}n}\,e^{-{\frac {3}{4}}n^{2}+{\frac {1}{12}}}}{G(n+1)}}}$

with the barnesian G function ${\displaystyle G(n+1)=1!\cdot 2!\cdot 3!\cdots (n-1)!.}$

Another possibility with the gamma function is: ${\displaystyle \Gamma }$

${\displaystyle A={\frac {2^{7/36}}{\pi ^{1/6}}}\exp \left({\frac {1}{3}}+{\frac {2}{3}}\int _{0}^{1/2}\!\ln \Gamma (x+1)\,\mathrm {d} x\right).}$

A series representation reads (Guillera, Sondow 2008)

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

literature

• Hermann Kinkelin : About a transcendent related to the gamma function and its application to the integral calculus . (July 1856) In: Journal for pure and applied mathematics , 57, 1860, pp. 122–138 (at GDZ: digizeitschriften.de )
• JWL Glaisher : On the Product 1¹.2².3³ ... nⁿ . In: The Messenger of Mathematics , 7, 1878, pp. 43–47 (English; “ A = 1 · 28242 7130” on p. 43); Text archive - Internet Archive

Web links

• Eric W. Weisstein : Glaisher-Kinkelin Constant . In: MathWorld (English).
• Sequence A087501 in OEIS ( continued fraction of A )

Individual evidence

1. ↑ 20,000 digits of the Glaisher-Kinkelin constant ( Memento of March 13, 2011 in the Internet Archive ) - the first 20,000 digits after the decimal point in the mpmath project (English)
2. ↑ Julian Havil: Gamma: Euler's constant, prime number beaches and the Riemann hypothesis . Springer-Verlag, Berlin 2007, ISBN 978-3-540-48495-0 , p. 103
3. ↑ Jesús Guillera, Jonathan Sondow: Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent . In: The Ramanujan Journal , 16, 2008, pp. 247–270 ( arxiv : math.NT / 0506319 )