Number of non-trivial zeros of $\zeta(s)$ in a small substrip I want to examine the number of non-trivial zeros of $\zeta(s)$ in the small substrip given by $a \leq \Re s \leq 1$, where $a>0$ is an absolute constant, and $U \leq \Im s \leq T$ by considering the following:
\begin{equation}\label{one} \tag{1}
N(a, T)-N(a, U),
\end{equation}
where $N(\sigma, t)$ denotes the number of non-trivial zeros of $\zeta(s)$, $\rho=\beta+i\gamma$, with $\sigma \leq \beta$ and $\gamma \leq t$. The issue is that, from what I've seen, all such formula for $N(\sigma, t)$ are given in the form $O(\cdots)$, which makes the study of \eqref{one} difficult, if not impossible. Does anyone have an suggestions or references to literature that may be of use?

It is known that there are on the order of $T \log T$ zeros up to height $T$ lying exactly on the critical line $\mathrm{Re} s = \tfrac{1}{2}$. It is possible to show that there are at most $O(T)$ zeros in the strip $\mathrm{Re} s \in (a, 1)$ for any $a > \tfrac{1}{2}$, so that one hundred percent of zeros lie arbitrarily close to the critical line.
There are more powerful versions of this result. I think a good reference to check would be Levinson's Almost all roots of $\zeta(s) = 0$ are arbitrarily close to $\sigma = 1/2$, appearing in the Proceedings of Nat. Acad. Sci. in 1975. (Link to paper).
Works citing Levinson's work will lead to a paper trail of similar analysis. (If you'll forgive a bit of self-promotion, I very recently wrote a note about related counting results, which I tried to make pretty straightforward).
The basic idea in these is to use some form of the argument principle to count zeros in regions and bounding the results. And the fundamental challenge is that we don't understand the actual distribution of zeros well enough to get anything except for upper bounds, as any actual zero would correspond to a main term in the integral analysis. There is no hope to perform this type of analysis without having results expressed as $O(\cdot)$ bounds with current ideas and techniques.
