Is there a well known result that states that as $t \to \infty$, 'almost all' zeros of any Dirichlet L function $L(s,\chi)$ lie in the region $R= \{\sigma+i t\mid |\sigma -\frac{1}{2}| \leq \Phi(t) \}$ for a positive function $\Phi$ "slowly going to zero" as $t\to \infty$, in the sense that the limit of the fraction of the # of zeros of the $L$ function, as $t \to \infty$, within $R\cap \{[0,1] \times[-t,t]\}$, is $1$?


In this case, precise quantifiers above would also be needed. I would be grateful for any possible reference (possibly there is a place in Iwaniec-Kowalski's book where this is mentioned precisely).



'Almost all' zeros of the Dirichlet L function lies 'near' the critical line?
One way to address this question is via a zero density result, of which there are a great variety. One can find a nice survey of this technology in Chapter 10 of Iwaniec and Kowalski. For instance, their Theorem 10.4 (which they call the Grand Density Theorem) gives a pretty general result that certainly works for an individual Dirichlet $L$-function with some control on the conductor. They also remark that work of Montgomery (*Topics in Multiplicative Number Theory*) can give some improvements which appear to be desirable based on the statement of the question.


