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

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.

Answer 1

This is correct and the relationship is most visible in Riemann's explicit formula for the weighted prime counting function. See for instance Garrett's write-up http://www.math.umn.edu/~garrett/m/mfms/notes_c/mfms_notes_02.pdf .

Answer 2

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.