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interestingly, <math>\frac{1}{\zeta(2)}</math> is also the natural density of the square-free numbers in {{math|'''N'''}}. In fact, this is not a coincidence. Consider the ''k''-dimensional lattice, <math>\mathbb{Z}^{k}</math>. The natural density of the points which are visible from the origin is <math>\frac{1}{\zeta(k)}</math>, which is also the natural density of the ''k''-th free integers in {{math|'''N'''}}.
===Divisor functions===
Consider the generalization of <math>d(n)</math>:
<math>\sigma_{\alpha}(n)=\sum_{d|n}d^{\alpha}</math>.
The following are true:
<math>
\sum_{n\le x}\sigma_{\alpha}(n)=
\begin{cases}
\;\;\sum_{n\le x}\sigma_{\alpha}(n)=\frac{\zeta(\alpha+1)}{\alpha+1}x^{\alpha+1}+O(x^{\beta}) \mbox{if } \alpha \mbox{ is positive} \\
\;\;\sum_{n\le x}\sigma_{-1}(n)=\zeta(2)x+O(logx) \mbox{if } \alpha=-1\\
\;\;\sum_{n\le x}\sigma_{\alpha}(n)=\zeta(-\alpha+1)x+O(x^{max(0,1+\alpha)}) \mbox{otherwise }
\end{cases}
</math>
where <math>\beta=max(1,\alpha)</math>.
==Better average order==
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