Sierpiński's constant

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)[1] denote the number of representations of

squares, then the Summatory Function[2] of

has the Asymptotic[3] expansion

+ π ln ⁡ n + o

The above plot shows

indicated as the solid horizontal line.

This article about a number is a stub.