View source for Sierpiński's constant

{{Short description|Mathematical constant}}
{{No footnotes|date=August 2012}}
'''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 [[Sum of squares function|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,
[[File:Sierpinski constant.gif|center|thumb|Graph of the given equation where the straight line represents Sierpiński's constant]]
Let r(n)<ref>{{Cite web|title=r(n)|url=https://archive.lib.msu.edu/crcmath/math/math/r/r004.htm|access-date=2021-11-30|website=archive.lib.msu.edu}}</ref> denote the number of representations of <math>n</math> by <math>k</math> squares, then the Summatory Function<ref>{{Cite web|title=Summatory Function|url=https://archive.lib.msu.edu/crcmath/math/math/s/s851.htm|access-date=2021-11-30|website=archive.lib.msu.edu}}</ref> of <math>r_2(k)/k</math> has the Asymptotic<ref>{{Cite web|title=Asymptotic|url=https://archive.lib.msu.edu/crcmath/math/math/a/a389.htm|access-date=2021-11-30|website=archive.lib.msu.edu}}</ref> 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 Sierpinski 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]]

== External links ==
* [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.
* {{MathWorld | urlname=SierpinskiConstant| title=Sierpinski Constant}}
* {{OEIS el|sequencenumber=A062089|name=Decimal expansion of Sierpiński's constant}}
*https://archive.lib.msu.edu/crcmath/math/math/s/s276.htm

{{DEFAULTSORT:Sierpinski's constant}}
[[Category:Mathematical constants]]

== References ==
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