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Sierpiński Constant

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SierpinskiConstant

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

sum_(k=1)^n(r_2(k))/k=pi(S+lnn)+O(n^(-1/2)),
(1)

where

S=gamma+(beta^'(1))/(beta(1))
(2)
=ln{(4pi^3e^(2gamma))/([Gamma(1/4)]^4)}
(3)
=(2.5849817595...)/pi
(4)
=0.8228252496...
(5)

(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 Function

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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 Constant

Cite this as:

Weisstein, Eric W. "Sierpiński Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SierpinskiConstant.html

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