Changes

* A sharper bound is

* A sharper bound is

::<math>\varphi(n) > \frac{n}{e^\gamma \log \log n + \frac{3}{\log \log n}}</math>

::<math>\varphi(n) > \frac{n}{e^\gamma \log \log n + \frac{3}{\log \log n}}</math>

* How fast does the totient function grow?  For large <math>n</math>, <math>\phi(n) \approx n</math> "on average".  One way to make this precise is by stating <math> \frac{1}{n^2} \sum_{k=1}^n \varphi(k) = \frac{3}{\pi^2}  + O \left( \frac{\log n}{n} \right)</math>, while in contrast <math> \frac{1}{n^2} \sum_{k=1}^n k = \frac{1}{2} + O \left( \frac{1}{n} \right).

* How fast does the totient function grow?  For large <math>n</math>, <math>\phi(n) \approx n</math> "on average".  One way to make this precise is by stating <math> \frac{1}{n^2} \sum_{k=1}^n \varphi(k) = \frac{3}{\pi^2}  + O \left( \frac{\log n}{n} \right)</math>, while in contrast <math> \frac{1}{n^2} \sum_{k=1}^n k = \frac{1}{2} + O \left( \frac{1}{n} \right).</math>

[[Category:Mathematics]]

[[Category:Mathematics]]