Precise relation between prime number theorem and zero-free region

I was wondering about the following, and I was hoping that some expert here could answer, rather than me indulging in a search for a needle in the haystack of formulas in books like Titchmarsch.

Notation:

• $zeta(s)$ is the Riemann zeta function.


• $f : mathbb R^+ rightarrow (0,1/2)$ is such that $zeta(s)$ does not vanish between $s = 1+it$ and $s=1 - f(t) + it$.


• $pi(x)$, $Li(x)$ as in wikipedia.

Assuming the above data, suppose the version of the prime number theorem that can be proven is:

$$ pi(x) = Li(x) + Oleft(G(x)right) $$

Question:

Can G(x) be given a closed form expression showing its precise(if and only if) dependence on $f(t)$?

Heuristics: When $f = 0$, $G(x) = x mathrm{e}^{-asqrt{ln x}}$ and when $f = 1/2$, $G(x) = sqrt x ln x$. So possibly there would be a term like $x^{1-f(x)}$ in a putative expression for $G(x)$.