Let $C(N)=\sum_{1<n\le N}{n^{-it}}$.
Vinogradov- Korobov estimate is
$$|C(N)| \le KN\exp\left(-\gamma \frac{\ln^3 N}{\ln^2 t}\right).$$
What are the best values of K and $\gamma$ ? I have only the values K=3 and $\gamma=1/49152$ from Ellison and Mendès-France book Les nombres premiers (1975)
Kevin Ford's paper "Vinogradov's integral and bounds for the Riemann zeta function" might have what you're looking for. Perhaps his Theorem 2 is related to what you want.