Pair Correlation Conjecture for the Zeros of the Riemann Zeta-function I: Simple and Critical Zeros

Abstract: Montgomery in 1973 introduced the Pair Correlation Conjecture (PCC) for zeros of the Riemann zeta-function. He also showed that a stronger conjecture would imply that asymptotically 100% of the zeros are simple. His reasoning to support these two conjectures made free use of the Riemann Hypothesis (RH). Building on Montgomery's approach, Gallagher and Mueller proved in 1978 that PCC under RH implies that 100% of the zeros are simple, but we show here that their method does not actually require RH. Thus Montgomery's second conjecture follows from his PCC conjecture. Recent work has shown that one can use pair correlation methods to obtain information not only on the vertical distribution of zeros, but also on the horizontal distribution. Applying this idea to Gallagher and Mueller's method, we show that PCC implies that asymptotically 100% of the zeros are both simple and on the critical line.