I want to find an approximation of Euler's totient.
It`s for product n of two prime number a and b
This is what I have actually: \begin{array}{} n = a.b\\ \varphi(n) = a.b - a - b + 1\\ \varphi(n) \approx n - 2.\sqrt{n}+1 \end{array} This work fine for small number, but with number up to 2**64, the error is small but not negligeable.
I want to scale to bigger number.
I search a better approximation or find an corrector term. Can you help me ?
Edit 1 : this post Upper bound for Euler's totient function on composite numbers show in this case (number with two prime factors) than my approximation is in reality an upper bound. After I read the book link by Will Jagy, I find this lower bound :
\begin{array}{} \frac{n}{\phi(n)} > {\rm e}^\gamma \log \log n\\ \frac{\phi(n)}{n} > \frac{1}{{\rm e}^\gamma \log \log n}\\ \phi(n) > \frac{1}{{\rm e}^\gamma \log \log n} . n\\ \end{array}
It exist a lower bound more accurate for my case ?