An Unconditional Montgomery Theorem and Simple Zeros of the Riemann Zeta-Function

Joint work with Siegfred Alan C. Baluyot, Daniel Alan Goldston, and Caroline L. Turnage-Butterbaugh


Assuming the Riemann Hypothesis (RH), Montgomery (1973) proved a theorem concerning the pair correlation of nontrivial zeros of the Riemann zeta-function. One consequence of this theorem was that, under RH, at least 2/3 of the zeros are simple. We show that this theorem of Montgomery holds unconditionally. As an application, under a much weaker hypothesis than RH, we show that at least 61.7% of zeros of the Riemann zeta-function are simple. This weaker hypothesis does not require that any of the zeros are on the half-line. We can further weaken the hypothesis using a density hypothesis.
Montgomery's theorem is a statement about the behavior of a certain function within the interval [-1,1] and it is conjectured to hold beyond that interval as well. This conjecture, assuming RH, implies that almost all zeros of the Riemann zeta-function are simple. As opposed to Montgomery's conjecture, the "Alternative Hypothesis"conjectures a completely different behavior of the function. If time allows, I would like to also briefly introduce related results under this Alternative Hypothesis.