Sierpiński's constant is a mathematical constant usually denoted as K. One way of defining it is as the following limit:
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) denote the number of representations of by squares, then the Summatory Function of has the Asymptotic expansion
,
where is the Sierpiński constant. The above plot shows
,
with the value of 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.
https://archive.lib.msu.edu/crcmath/math/math/s/s276.htm
References