Rahul Sharma (Editor)

Sierpiński's constant

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

Sierpiński's constant is a mathematical constant usually denoted as K. One way of defining it is by limiting the expression:

K = lim n → ∞ [ ∑ k = 1 n r 2 ( k ) k − π ln ⁡ n ]

where r2(k) is a number of representations of k as a sum of the form a2 + b2 for natural a and b.

It can be given in closed form as:

K = π ( 2 ln ⁡ 2 + 3 ln ⁡ π + 2 γ − 4 ln ⁡ Γ ( 1 4 ) ) = π ln ⁡ ( 4 π 3 e 2 γ Γ ( 1 4 ) 4 ) = π ln ⁡ ( e 2 γ 2 G 2 ) = 2.584981759579253217065893587383 …

where G is Gauss's constant and γ is the Euler-Mascheroni constant.

References

Sierpiński's constant Wikipedia
(Text) CC BY-SA