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.