Sierpiński's constant

Sierpiński's constant is a mathematical constant usually denoted as K. One way of defining it is by limiting the expression:

K=\lim _{n\to \infty }\left[\sum _{k=1}^{n}{r_{2}(k) \over k}-\pi \ln n\right]

where r₂(k) is a number of representations of k as a sum of the form a² + b² for natural a and b.

It can be given in closed form as:

{\begin{aligned}K&=\pi \left(2\ln 2+3\ln \pi +2\gamma -4\ln \Gamma \left({\tfrac {1}{4}}\right)\right)\\&=\pi \ln \left({\frac {4\pi ^{3}e^{2\gamma }}{\Gamma \left({\tfrac {1}{4}}\right)^{4}}}\right)\\&=\pi \ln \left({\frac {e^{2\gamma }}{2G^{2}}}\right)\\&=2.584981759579253217065893587383\dots \end{aligned}}

where $G$ is Gauss's constant and $\gamma$ is the Euler-Mascheroni constant.

See also

Wacław Sierpiński

External links

http://www.scenta.co.uk/tcaep/science/constant/details/sierpinskisconstant.xml
http://www.plouffe.fr/simon/constants/sierpinski.txt - Sierpiński's constant up to 2000th decimal digit.
Weisstein, Eric W. "Sierpinski Constant". MathWorld.

"Sloane's A062089". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

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