Effective bounds on Euler's totient - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T21:56:27Z http://mathoverflow.net/feeds/question/65390 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65390/effective-bounds-on-eulers-totient Effective bounds on Euler's totient Charles 2011-05-19T01:49:55Z 2011-05-19T02:01:09Z <p>Quick question: It's known that $$\limsup\frac{n}{\varphi(n)\log\log n}=e^\gamma$$ but are there known C and N such that $$\varphi(n)>\frac{Cn}{e^\gamma\log\log n}$$ for all $n>N$?</p> <p>Failing that, what are good effective bounds on $\varphi$? The square root bound isn't good enough for me.</p> http://mathoverflow.net/questions/65390/effective-bounds-on-eulers-totient/65391#65391 Answer by Igor Rivin for Effective bounds on Euler's totient Igor Rivin 2011-05-19T01:58:07Z 2011-05-19T02:01:09Z <p>Yes. Look at http://en.wikipedia.org/wiki/Euler's_totient_function#Inequalities:</p> <p>$$\varphi(n)>\frac{n}{e^\gamma\log\log n + \frac{3}{\log\log n}}$$ for $n>2$.</p>