MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

Your heuristic is wrong: $G(x)=x\exp{(-a\sqrt{\log{x}}})$ follows from $f=\frac{c}{\log{(|t|+3)}}$ for some fixed real $c>0$.
$\zeta'(s)/\zeta(s)$ 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$so that the same book)two error terms balance. 1 Your heuristic is wrong:$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).