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* The [[prime number theorem]] is equivalent to the statement that the [[von Mangoldt function]] Λ(''n'') has average order 1;
* An average order of μ(''n''), the [[Möbius function]], is zero; this is again equivalent to the [[prime number theorem]].
===Visibility of lattice points===
We say that two lattice points are visible from one another if there is no lattice point on the open line segment joining them.
Now, if ''gcd(a, b)=d>1'', then writing ''a=da’'', ''b=db’'' one observes that the point
''(a’, b’)'' is on the line segment which joins ''(0,0)'' to ''(a, b)'' and hence ''(a, b)'' is not visible from the origin. Thus ''(a, b)'' is visible from the origin implies that ''(a, b)=1''. Conversely, it is also easy to see that ''gcd(a, b)=1'' implies that there is no other integer lattice point in the segment joining ''(0,0)'' to ''(a, b)''.
Thus, ''(a, b)'' is visible from ''(0,0)'' if and only if gcd(a, b)=1.
Notice that <math>\frac{\varphi(n)}{n}</math> is the probability of a random point on the square <math>\{(r,s)\in \mathbb{N} : \max(|r|,|s|)=n\}</math> to be visible from the origin.
Thus, one can show that the natural density of the points which are visible from the origin is given by the average,
<math>\lim_{N\rightarrow \infty}\frac{1}{N}\sum_{n\le N}\frac{\varphi(n)}{n}=\frac{6}{\pi^{2}}=\frac{1}{\zeta(2)}</math>.
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'''}}.
==Better average order==
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