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.

It can be given in closed form as:

K=\pi \left(2 \ln 2+3 \ln \pi + 2 \gamma - 4 \ln \Gamma \left(\frac{1}{4}\right)\right)\approx 2.58498 17595 79253 21706 58935 87383\dots

See also

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