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Sierpiński's constant
Mathematical constant From Wikipedia, the free encyclopedia
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Sierpiński's constant is a mathematical constant usually denoted as K. One way of defining it is as the following limit:
![]() | This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. (August 2012) |
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:
where is the lemniscate constant and is the Euler-Mascheroni constant.
Another way to define/understand Sierpiński's constant is,

Let r(n)[1] denote the number of representations of by squares, then the Summatory Function[2] of has the Asymptotic[3] expansion
,
where is the Sierpiński constant. The above plot shows
,
with the value of indicated as the solid horizontal line.
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See also
External links
- http://www.plouffe.fr/simon/constants/sierpinski.txt - Sierpiński's constant up to 2000th decimal digit.
- Weisstein, Eric W. "Sierpinski Constant". MathWorld.
- OEIS sequence A062089 (Decimal expansion of Sierpiński's constant)
- https://archive.lib.msu.edu/crcmath/math/math/s/s276.htm
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References
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