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.

Its value is:

$K=\pi \left(2 \ln 2%2B3 \ln \pi %2B 2 \gamma - 4 \ln \Gamma \left(\frac{1}{4}\right)\right)\approx 2.58498 17595 79253 21706 58935 87383\dots$

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