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 → ∞ [ ∑ k = 1 n r 2 ( k ) k − π ln ⁡ n ] {\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:

K = π ( 2 ln ⁡ 2 + 3 ln ⁡ π + 2 γ − 4 ln ⁡ Γ ( 1 4 ) ) = π ln ⁡ ( 4 π 3 e 2 γ Γ ( 1 4 ) 4 ) = π ln ⁡ ( π 2 e 2 γ 2 ϖ 2 ) = 2.584981759579253217065893587383 … {\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 ϖ {\displaystyle \varpi } is the lemniscate constant and γ {\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 n {\displaystyle n} by k {\displaystyle k} squares, then the Summatory Function of r 2 ( k ) / k {\displaystyle r_{2}(k)/k} has the Asymptotic expansion

∑ k = 1 n r 2 ( k ) k = K + π ln ⁡ n + o ( 1 n ) {\displaystyle \sum _{k=1}^{n}{r_{2}(k) \over k}=K+\pi \ln n+o\!\left({\frac {1}{\sqrt {n}}}\right)} , where K = 2.5849817596 {\displaystyle K=2.5849817596} is the Sierpiński constant. The above plot shows

( ∑ k = 1 n r 2 ( k ) k ) − π ln ⁡ n {\displaystyle \left(\sum _{k=1}^{n}{r_{2}(k) \over k}\right)-\pi \ln n} , with the value of K {\displaystyle K} indicated as the solid horizontal line.

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