Hello,


I am reading some material on circle method. Right now
I am at its application to the binary Goldbach problem.
To obtain a certain bound the fact
$\sum\_{n> X} 1/ \phi(n)^2 = O(1/X)$ 
is used. Could anyone please help me how to solve it?


Here $\phi$ is the Euler totient function.
And I would like the result without $\epsilon$.


Thank you.



sum of $1/ \phi(n)^2$
Theorem 2.14 of "Multiplicative Number Theory" I. Classical Theory, by Montgomery & Vaughan
implies that 
$$\sum\_{n\leq x} \left(\frac{n}{\phi(n)}\right)^2 = O(x)$$


Use this and partial summation method with 
$$\sum\_{n\leq x} \left(\frac{n}{\phi(n)}\right)^2 \frac{1}{n^2}$$


