Sierpiński's constant

Sierpiński's constant is a mathematical constant usually denoted as K. One way of defining it is as the following limit:


 * $$K=\lim_{n \to \infty}\left[\sum_{k=1}^{n}{r_2(k)\over k} - \pi\ln n\right]$$

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

Another way to define/understand Sierpiński's constant is, Let r(n) denote the number of representations of $$n$$ by $$k$$ squares, then the Summatory Function of $$r_2(k)/k$$ has the Asymptotic expansion

$$\sum_{k=1}^{n}{r_2(k)\over k}=K+\pi\ln n+o\surd(1/n)$$,

where $$ K=2.5849817596$$ is the Sierpinski constant. The above plot shows

$$[\sum_{k=1}^{n}{r_2(k)\over k}]-\pi\ln n$$,

with the value of $$K$$ indicated as the solid horizontal line.