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Rencontres de théorie analytique des nombres

An unconditional Montgomery theorem and simple zeros of the Riemann Zeta function

Salle Grisvard, IHP, Paris

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.

This is a joint work with Siegfred Alan C. Baluyot, Daniel Alan Goldston, and Caroline L. Turnage-Butterbaugh.