Number of non-trivial zeros of $\zeta(s)$ in a small substrip

Question

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?

Answer 1

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.

Answer 2

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.