Sierpiński's constant

${\displaystyle K=\underset{n\to \mathrm{\infty }}{lim}\left[\sum _{k=1}^n\frac{r_2(k)}{k}-\pi \ln n\right]}$

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:

${\displaystyle \begin{array}{cc}K& =\pi \left(2\ln 2+3\ln \pi +2\gamma -4\ln \mathrm{\Gamma }\left(\frac{1}{4}\right)\right)\\ & =\pi \ln \left(\frac{4{\pi }^3{e}^{2\gamma }}{\mathrm{\Gamma }{\left(\frac{1}{4}\right)}^4}\right)\\ & =\pi \ln \left(\frac{{\pi }^2{e}^{2\gamma }}{2{\varpi }^2}\right)\\ & =2.584981759579253217065893587383\dots \end{array}}$

where ${\displaystyle \varpi }$ is the lemniscate constant and ${\displaystyle \gamma }$ is the Euler-Mascheroni constant.

Another way to define/understand Sierpiński's constant is,

Let r(n) denote the number of representations of ${\displaystyle n}$ by ${\displaystyle k}$ squares, then the Summatory Function of ${\displaystyle r_2(k)/k}$ has the Asymptotic expansion

${\displaystyle \sum _{k=1}^n\frac{r_2(k)}{k}=K+\pi \ln n+o\left(\frac{1}{\sqrt{n}}\right)}$,

where ${\displaystyle K=2.5849817596}$ is the Sierpinski constant. The above plot shows

${\displaystyle \left(\sum _{k=1}^n\frac{r_2(k)}{k}\right)-\pi \ln n}$,

with the value of ${\displaystyle K}$ indicated as the solid horizontal line.