The Sierpiński constant 
is defined by the asymptotic
expansion of the summatory function of 
, where the sum
of squares function 
counts ordered pairs
of integers whose squares
sum to 
.
As 
,
 |
(1)
|
where
 |  |  |
(2)
|
 |  | ![ln{(4pi^3e^(2gamma))/([Gamma(1/4)]^4)}](/images/equations/SierpinskiConstant/Inline11.svg) |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
(OEIS A241017; Finch 2003, p. 123), 
is the Dirichlet
beta function, 
is the Euler-Mascheroni constant, and
is the gamma function. The alternative normalization
(OEIS A062089) is also called the Sierpiński
constant.
