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 conjectured that asymptotically 100% of the zeros are simple. His reasoning to support these two conjectures used 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. Actually, the method of Gallagher and Mueller does not depend on RH, and thus Montgomery's second simplicity conjecture follows unconditionally from his PCC conjecture. We clarify this result by explicitly not assuming RH and considering PCC as a conjecture only concerning the vertical distribution of zeros. We then show that, for the first time, PCC can also be used to obtain information on the horizontal distribution of zeros. Using Gallagher and Mueller's method and a new idea concerning "horizontal multiplicity", we use PCC to prove that asymptotically 100% of the zeros are not only simple but also on the critical line.

Theorems & Definitions (12)

• Theorem 1
• Remark 1
• Proposition 1: Gallagher and Mueller GaMu78
• Proposition 2: Fujii
• Remark 2
• Remark 3
• Theorem 2
• proof : Proof of Theorem \ref{['thm2']}
• Remark 4
• proof : Proof of Theorem \ref{['thm1']}