Sierpiński's constant

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

$K=\underset{n\to \mathrm{\infty }}{\lim }[\sum _{k=1}^n\frac{r_2(k)}{k}-\pi \ln n]$

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:

$\begin{array}{rl}K& =\pi (2\ln 2+3\ln \pi +2\gamma -4\ln \mathrm{\Gamma }\left(\frac{1}{4}\right))\\ & =\pi \ln \left(\frac{4{\pi }^3{e}^{2\gamma }}{\mathrm{\Gamma }{\left(\frac{1}{4}\right)}^4}\right)\\ & =\pi \ln \left(\frac{e^{2\gamma }}{2G^2}\right)\\ & =2.584981759579253217065893587383\dots \end{array}$

where $G$ is Gauss's constant and $\gamma$ is the Euler-Mascheroni constant.

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

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Graph of the given equation where the straight line represents Sierpiński's constant.

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

$\sum _{k=1}^n\frac{r_2(k)}{k}=K+\pi \ln n+o\surd (1/n)$,

where $K=2.5849817596$ is the Sierpinski constant. The above plot shows

$[\sum _{k=1}^n\frac{r_2(k)}{k}]-\pi \ln n$,

with the value of $K$ indicated as the solid horizontal line.

See also

• Wacław Sierpiński

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