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Your heuristic is wrong: $G(x)=x\exp{(-a\sqrt{\log{x}}})$ follows from $f=\frac{c}{\log{(|t|+3)}}$ for some fixed real $c>0$.

I really don't want to tell you the answer, because this is a great exercise! A big hint: use equation (5.28) in Iwaniec and Kowalski, to convert your zero-free region to an upper bound for the "approximate explicit formula"

$\zeta'(s)/\zeta(s)$ along \psi(x)=x-\sum_{|\rho|\leq T} \frac{x^{\rho}}{\rho}+O(T^{-1} x \log^2{x}),$

bound the edge of your region, sum over zeros trivially given what you know about $f$, and then use the approximate version of Perron's formula (equation (5.111) in choose $T$ so that the same book)two error terms balance.
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Your heuristic is wrong: $G(x)=x\exp{(-a\sqrt{\log{x}}})$ follows from $f=\frac{c}{\log{(|t|+3)}}$ for some fixed real $c>0$.

I really don't want to tell you the answer, because this is a great exercise! A big hint: use equation (5.28) in Iwaniec and Kowalski, to convert your zero-free region to an upper bound for $\zeta'(s)/\zeta(s)$ along the edge of your region, and then use the approximate version of Perron's formula (equation (5.111) in the same book).