Sierpiński's constant is a mathematical constant usually denoted as K. One way of defining it is as the following limit:
\( {\displaystyle K=\lim _{n\to \infty }\left[\sum _{k=1}^{n}{r_{2}(k) \over k}-\pi \ln n\right]} \)
where \( r_2(k) \) is a number of representations of k as a sum of the form \( a^2 + b^2 \) for integer a and b.
It can be given in closed form as:
\( { {\displaystyle {\begin{aligned}K&=\pi \left(2\ln 2+3\ln \pi +2\gamma -4\ln \Gamma \left({\tfrac {1}{4}}\right)\right)\\&=\pi \ln \left({\frac {4\pi ^{3}e^{2\gamma }}{\Gamma \left({\tfrac {1}{4}}\right)^{4}}}\right)\\&=\pi \ln \left({\frac {e^{2\gamma }}{2G^{2}}}\right)\\&=2.584981759579253217065893587383\dots \end{aligned}}} \)
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[permanent dead link]
http://www.plouffe.fr/simon/constants/sierpinski.txt - Sierpiński's constant up to 2000th decimal digit.
Weisstein, Eric W. "Sierpinski Constant". MathWorld.
OEIS sequence A062089 (Decimal expansion of Sierpiński's constant)
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
