Sierpinski Constant

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

Let r(n) denote the number of representations of by squares, then the Summatory Function of 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

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

indicated as the solid horizontal line.

See also r(n)

References

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