Pair Correlation Conjecture for the zeros of the Riemann zeta-function II: The Alternative Hypothesis

2025/07/09 by Goldston, Daniel A., Lee, Junghun, Schettler, Jordan +1 · 1 citation
#11M06 #11M26 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.06823

Abstract

In an earlier paper, we proved that Montgomery's Pair Correlation Conjecture (PCC) for zeros of the Riemann zeta-function can be used to prove without the assumption of the Riemann Hypothesis (RH) that asymptotically 100% of the zeros are both simple and on the critical line. This is based on a method of Gallagher and Mueller from 1978. We formulate an appropriate form of the Alternative Hypothesis (AH), which determines a different PCC, and, using the same method as above, prove that asymptotically, 100% of the zeros are both simple and on the critical line. As in our previous paper, we do not assume RH.

Citations

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

Cited by

• Zeta Zeros on the Critical Line

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