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,
Graph of the given equation where the straight line represents Sierpiński's constant.
Let r(n) denote the number of representations of by squares, then the Summatory Function of has the Asymptotic expansion
where is the Sierpinski constant. The above plot shows
with the value of indicated as the solid horizontal line.