Sierpiński's constant

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 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:

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

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

${\displaystyle \sum _{k=1}^{n}{r_{2}(k) \over k}=K+\pi \ln n+o\!\left({\frac {1}{\sqrt {n}}}\right)}$,

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

${\displaystyle \left(\sum _{k=1}^{n}{r_{2}(k) \over k}\right)-\pi \ln n}$,

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