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 Gauss's 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.