Suppose I have an integral Dirichlet series $f(s) = \sum c_n n^{-s}$, $c_n \in \mathbb{Z}$, with at least one non-zero term $c_N$. Suppose furthermore that this series converges absolutely and uniformly for $\mathrm{Re}(s) > 1 + \delta$ for any $\delta > 0$ (so that $c_n$ grows slower than $n^\epsilon$ for any $\epsilon$).
I want $f(s)$ to be "small" (in terms of $N$) for fixed $\mathrm{Re}(s)$ and a range of $\mathrm{Im}(s)$. How small can I get it over any given range?
I think one can take sums of $(N + \Delta n)^{-s}$ for $\Delta n \approx O(\ln N)$ and cancel out terms in the Taylor series up to order $N^{-(s + O(\ln N))}$, all while keeping the coefficients polylog in N for $Im(s) \approx O(\ln N)$. Can we do any better? What about larger values of $s$?