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Sierpinski Constant

\begin{figure}\begin{center}\BoxedEPSF{SierpinskiConstant.epsf}\end{center}\end{figure}

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

\begin{displaymath}
\sum_{k=1}^n {r_2(k)\over k}=K+\pi\ln n+{\mathcal O}(n^{-1/2}),
\end{displaymath}

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

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

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

See also r(n)


References

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




© 1996-9 Eric W. Weisstein
1999-05-26