Sierpiński Constant
Let the sum of squares function 👁 r_k(n)
denote the number of representations of 👁 n
by 👁 k
squares, then the summatory
function of 👁 r_2(k)/k
has the asymptotic expansion
where
| 👁 S | 👁 = | 👁 gamma+(beta^'(1))/(beta(1)) |
(2)
|
| 👁 Image | 👁 = | 👁 ln{(4pi^3e^(2gamma))/([Gamma(1/4)]^4)} |
(3)
|
| 👁 Image | 👁 = | 👁 (2.5849817595...)/pi |
(4)
|
| 👁 Image | 👁 = | 👁 0.8228252496... |
(5)
|
![👁 ln{(4pi^3e^(2gamma))/([Gamma(1/4)]^4)}](https://mathworld.wolfram.com/images/equations/SierpinskiConstant/Inline10.svg){kind=link}
(OEIS A241017) is the Sierpiński constant (Finch 2003, p. 123), 👁 beta(x)
is the Dirichlet
beta function, 👁 gamma
is the Euler-Mascheroni constant, and
👁 Gamma(x)
is the gamma function.
See also
Sum of Squares FunctionExplore with Wolfram|Alpha
More things to try:
References
Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 114, 2003.Sierpiński, W. Oeuvres Choisies, Tome 1. Editions Scientifiques de Pologne, 1974.Sloane, N. J. A. Sequences A062089 and A241017 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Sierpiński ConstantCite this as:
Weisstein, Eric W. "Sierpiński Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SierpinskiConstant.html