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:

$\begin{align} K &= \pi \left(2 \ln 2+3 \ln \pi + 2 \gamma - 4 \ln \Gamma \left(\tfrac{1}{4}\right)\right)\\ &=\pi \ln\left(\frac{4\pi^3 e^{2\gamma}}{\Gamma \left(\tfrac{1}{4}\right)^4}\right)\\ &=\pi \ln\left(\frac{e^{2\gamma}}{2G^2}\right)\\ &= 2.58498 17595 79253 21706 58935 87383\dots \end{align}$

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.