Number/Representation Theory

H. L. Montgomery (1973) suggested an approach to study the pair correlation of nontrivial zeros of the Riemann zeta-function, and proved the corresponding asymptotic formula within a limited range assuming the Riemann Hypothesis (RH). The extended behavior remains a conjecture which implies the famous Pair Correlation Conjecture (PCC) for these zeros. In recent work with Siegfred Alan C. Baluyot, Daniel Alan Goldston, and Caroline L. Turnage-Butterbaugh, we showed how to remove RH in Montgomery's pair correlation method and recover known results on the proportion of simple zeros under hypotheses weaker than RH. Since we do not assume RH, we can now deal with possible zeros off the critical line which then shows that the zeros captured are not only simple but also on the critical line. In separate work with Daniel Goldston, Junghun Lee and Jordan Schettler, we showed that PCC or another pair correlation conjecture, without RH, implies that asymptotically 100% of the zeros are simple and on the critical line, thus RH and the Simple Zero Conjecture are asymptotically true.

We have recently introduced a new notion called ``horizontal multiplicity" of zeros which is exactly the usual multiplicity if RH is true. Without RH, this clarifies possible deviations of the real part distribution of the zeros and helps us demonstrate our ideas better. One main idea to remove the RH assumption is to take advantage of the feature that all zeros off the critical line come in pairs on the same horizontal line. With Daniel Goldston, we clarified our ideas using very simple elementary arguments, and this is what I aim to introduce in this talk.