Pair Correlation of Zeros of the Riemann Zeta Function I: Proportions of Simple Zeros and Critical Zeros

Abstract: Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem in 1973 concerning the pair correlation of zeros of the Riemann zeta-function and applied this to prove that at least $2/3$ of the zeros are simple. In earlier work we showed how to remove RH from Montgomery's theorem and, in turn, obtain results on simple zeros assuming conditions on the zeros that are weaker than RH. Here we assume a more general condition, namely that all the zeros $\rho = \beta +i\gamma$ with $T<\gamma\le 2T$ are in a narrow vertical box centered on the critical line with width $\frac{b}{\log T}$. For simplicity, now assume that $b\to 0$ as $T\to \infty$. Following Montgomery's method, we prove the generalization of Montgomery's result that at least $2/3$ of zeros are simple, and also the new result that at least $2/3$ of the zeros are on the critical line. We also use the pair correlation method to prove that at least $1/3$ of the zeros are both simple and on the critical line, a result already known unconditionally. Our work thus shows that the pair correlation method can be used to prove results not only on the vertical distribution of zeros but also on their horizontal distribution.