Zeta Zeros in a Narrow Vertical Box

TL;DR

This paper proves that if all zeros of the Riemann zeta-function are confined within a narrow vertical box near the critical line, then at least two-thirds of these zeros are simple and on the line, assuming the zeros are in that box.

Contribution

It provides a simple proof that zeros in a narrow vertical region near the critical line are mostly simple and on the line, generalizing Montgomery's approach.

Findings

At least 2/3 of zeros are simple and on the critical line within the narrow box.

The proof assumes zeros are confined in a vertical box with width shrinking as T increases.

The result extends Montgomery's original proof under a new geometric assumption.

Abstract

In 1973 Montgomery proved, assuming the Riemann Hypothesis (RH), that asymptotically at least 2/3 of zeros of the Riemann zeta-function are simple zeros. In a previous note (arXiv:2511.20059 [math.NT]) we showed how RH can be replaced with a general estimate for a double sum over zeros, and this allows one to then obtain results on zeros that are both simple and on the critical line. Here we give a simple proof based on a direct generalization of Montgomery's proof that on assuming all the zeros are in a narrow vertical box between height $T$ and $2T$ of width $b/\log T$ and centered on the critical line, then, if $b=b(T)\to 0$ as $T\to \infty$, we have asymptotically at least 2/3 of the zeros are simple and on the critical line.