Sierpiński's constant
Sierpiński's constant is a mathematical constant usually denoted as K. One way of defining it is as the following limit:
- [math]\displaystyle{ K=\lim_{n \to \infty}\left[\sum_{k=1}^{n}{r_2(k)\over k} - \pi\ln n\right] }[/math]
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:
- [math]\displaystyle{ \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{e^{2\gamma}}{2G^2}\right)\\ &= 2.58498 17595 79253 21706 58935 87383\dots \end{align} }[/math]
where [math]\displaystyle{ G }[/math] is Gauss's constant and [math]\displaystyle{ \gamma }[/math] is the Euler-Mascheroni constant.
Another way to define/understand Sierpiński's constant is,
Let r(n)[1] denote the number of representations of [math]\displaystyle{ n }[/math] by [math]\displaystyle{ k }[/math] squares, then the Summatory Function[2] of [math]\displaystyle{ r_2(k)/k }[/math] has the Asymptotic[3] expansion
[math]\displaystyle{ \sum_{k=1}^{n}{r_2(k)\over k}=K+\pi\ln n+o\surd(1/n) }[/math],
where [math]\displaystyle{ K=2.5849817596 }[/math] is the Sierpinski constant. The above plot shows
[math]\displaystyle{ [\sum_{k=1}^{n}{r_2(k)\over k}]-\pi\ln n }[/math],
with the value of [math]\displaystyle{ K }[/math] indicated as the solid horizontal line.
See also
External links
- [1]
- http://www.plouffe.fr/simon/constants/sierpinski.txt - Sierpiński's constant up to 2000th decimal digit.
- Weisstein, Eric W.. "Sierpinski Constant". http://mathworld.wolfram.com/SierpinskiConstant.html.
- OEIS sequence A062089 (Decimal expansion of Sierpiński's constant)
- https://archive.lib.msu.edu/crcmath/math/math/s/s276.htm
References
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