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Sierpinski Constant


Let r(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)\over k}=K+\pi\ln n+{\mathcal O}(n^{-1/2}),

where $K=2.5849817596$ is the Sierpinski constant. The above plot shows

\left[{\sum_{k=1}^n {r_2(k)\over k}}\right]-\pi\ln n,

with the value of $K$ indicated as the solid horizontal line.

See also r(n)


Sierpinski, W. Oeuvres Choisies, Tome 1. Editions Scientifiques de Pologne, 1974.

© 1996-9 Eric W. Weisstein