An unconditional Montgomery theorem for pair correlation of zeros of the Riemann zeta-function
Streszczenie
Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem concerning pair correlation of zeros of the Riemann zeta-function. One consequence of this theorem is that, assuming RH, at least $67.9\%$ of the nontrivial zeros are simple. Here we obtain an unconditional form of Montgomery’s theorem and show how to apply it to prove the following result on simple zeros: If all the zeros $\rho =\beta +i\gamma $ of the Riemann zeta-function such that $T^{3/8} \lt \gamma \le T$ satisfy $|\beta -1/2| \lt 1/(2\log T)$, then, as $T$ tends to infinity, at least $61.7\%$ of these zeros are simple. The method of proof neither requires nor provides any information on whether any of these zeros are or are not on the critical line where $\beta =1/2$. We also obtain the same result under the weaker assumption of a strong zero-density hypothesis.
