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?

Firstly, it should come as no surprise that results on $N(\sigma, T)$ only give bounds. After all, any other kind of approximation would disprove RH.
I would not consider myself an expert in the field, but it seems to me as if any kind of formula for $N(\sigma, T)$ has its origins in zero detecting devices, which usually refers to some kind of necessary condition for some complex number $\rho = s + it$ (with $s > \frac 12$ and $t > 0$) to be a zero of $\zeta$. For example, for $X > 0$ we might write
$$ M_X(s) = \sum_{n \leq X} \frac{\mu(n)}{n^s},$$
approximating the inverse of $\zeta$ given by
$$\zeta(s)^{-1} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s}.$$
The idea is now to use the equation $\zeta(s) M_\infty(s) = 1$ to deduce an approximation
$$\zeta(s) M_X(s) = 1 + R(s)$$
with some (usually) small error $R$, which if inserted some root $\rho$ yields
$$0 = 1 + R(\rho), \quad \text{i.e.} \quad R(\rho) \gg 1,$$
showing that the usually small error is somehow kinda not so small when inserted a root.
You can now use purely analytic methods to bound how often we can have $R(\rho) \gg 1$. Usually results are given for $T/2 < \Im \rho < T$, but you might try to slightly change the argument to obtain results for
$U < \Im \rho < T$. I think the book "The Riemann Zeta-Function" by Karatsuba and Voronin is a great reference for these arguments.
