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=\lim_\left sum_^ - \pi\ln n\right/math>

where ''r''₂(''k'') is a number of representations of ''k'' as a sum of the form ''a''² + ''b''² for integer ''a'' and ''b''.

It can be given in closed form as:
:$\begin K &= \pi \left(2 \ln 2+3 \ln \pi + 2 \gamma - 4 \ln \Gamma \left(\tfrac\right)\right)\\ &=\pi \ln\left(\frac\right)\\ &=\pi \ln\left(\frac\right)\\ &= 2.58498 17595 79253 21706 58935 87383\dots \end$

where $\varpi$ is the lemniscate constant and $\gamma$ is the Euler-Mascheroni constant.

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

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

$\sum_^=K+\pi\ln n+o\!\left(\frac\right)$,

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

$\left(\sum_^\right)-\pi\ln n$,

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

See also

* Wacław Sierpiński

External links

* http://www.plouffe.fr/simon/constants/sierpinski.txt - Sierpiński's constant up to 2000th decimal digit.
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*https://archive.lib.msu.edu/crcmath/math/math/s/s276.htm

Mathematical constants

References

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