I am considering a function related to the Riemann zeta function, and I have a asymptotic for the cardinality of the zeros less than $T$, of the form $$N(T) = cT^2 + O(T)$$ when $T$ is large. Apparently it is possible from this "global" information the "local" fact that there are few zeros in small balls, precisely that the number of zeros $\rho$ such that $|s - \rho| < 2$ is $O(Im(|s|))$.
I do not see any reason for this to be true (at least for a general function, zeros can be globally few while concentrating in specific small balls). Where the specific structure of the Riemann zeta function comes into play? Are there precise references for it?