TITLE: Riemann Hypothesis: Is $1/2$ of critical line same as the $1/2$ of square-root accurately of error term of prime number theorem?
QUESTION [7 upvotes]: Here is a question about Riemann Hypothesis: 
Is $1/2$ of critical line same as the $1/2$ of square-root accurately of error term of prime number theorem $?$
In other words, (just for some brain exercise), let's assume that critical line is at $x=3/4$, does this mean that the error term in prime number theorem will be proportional to the power of $3/4$ $?$ So $|\pi(x) - Li(x) |$ would be bounded by $x^{(3/4)}$ $?$
Which book has a simple proof of this $?$
Thank you.

REPLY [3 votes]: Yes.
Ingham, The Distribution of Prime Numbers, contains a proof that the lower bound $\Theta_1$ of numbers $\alpha$ for which it is true that
$$\psi(x)-x = O(x^{\alpha}) $$ 
is equal to the upper bound $\Theta_2$ of the real parts of the zeros of $\zeta(s).$
This is his Theorem 31 (with a proof) on page 84.