Sierpiński Constant

Let the sum of squares function  denote the number of representations of  by  squares, then the summatory function of  has the asymptotic expansion

(1)

where

			(2)
			(3)
			(4)
			(5)

(OEIS A241017) is the Sierpiński constant (Finch 2003, p. 123),  is the Dirichlet beta function,  is the Euler-Mascheroni constant, and  is the gamma function.

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."