Sierpiński's constant is a mathematical constant usually denoted as K. One way of defining it is as the following limit:

:<math>K=\lim_{n \to \infty}\left[\sum_{k=1}^{n}{r_2(k)\over k} - \pi\ln n\right]</math>

where r<sub>2</sub>(k) is a number of representations of k as a sum of the form a<sup>2</sup>&nbsp;+&nbsp;b<sup>2</sup> for integer a and b.

It can be given in closed form as:

:<math>\begin{align}

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^2e^{2\gamma{2\varpi^2}\right)\\

&= 2.58498 17595 79253 21706 58935 87383\dots

\end{align}</math>

where <math>\varpi</math> is the lemniscate constant and <math>\gamma</math> is the Euler-Mascheroni constant.

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

center|thumb|Graph of the given equation where the straight line represents Sierpiński's constant

Let r(n) denote the number of representations of <math>n</math> by <math>k</math> squares, then the Summatory Function of <math>r_2(k)/k</math> has the Asymptotic expansion

<math>\sum_{k=1}^{n}{r_2(k)\over k}=K+\pi\ln n+o\!\left(\frac{1}{\sqrt n}\right)</math>,

where <math> K=2.5849817596</math> is the Sierpiński constant. The above plot shows

<math>\left(\sum_{k=1}^{n}{r_2(k)\over k}\right)-\pi\ln n</math>,

with the value of <math>K</math> indicated as the solid horizontal line.

See also

  • Wacław Sierpiński
  • [https://web.archive.org/web/20070927080657/http://www.scenta.co.uk/tcaep/science/constant/details/sierpinskisconstant.xml]
  • http://www.plouffe.fr/simon/constants/sierpinski.txt - Sierpiński's constant up to 2000th decimal digit.
  • https://archive.lib.msu.edu/crcmath/math/math/s/s276.htm

References