Sierpiński's constant is a mathematical constant usually denoted as K. One way of defining it is as the following limit:
![{\displaystyle K=\lim _{n\to \infty }\left[\sum _{k=1}^{n}{r_{2}(k) \over k}-\pi \ln n\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2d934ded4d1e09bee67d6a314ee18d2c80f1b3)
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 Sierpinski constant. The above plot shows
,
with the value of indicated as the solid horizontal line.
See also
[edit]External links
[edit]- [1]
- 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
References
[edit]- ^ "r(n)". archive.lib.msu.edu. Retrieved 2021-11-30.
- ^ "Summatory Function". archive.lib.msu.edu. Retrieved 2021-11-30.
- ^ "Asymptotic". archive.lib.msu.edu. Retrieved 2021-11-30.