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arXiv · 2507.06823

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

Daniel A. Goldston·Junghun Lee·Jordan Schettler·Ade Irma Suriajaya
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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.

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Daniel A. Goldston, Junghun Lee, Jordan Schettler, Ade Irma Suriajaya. 2025-07-09. Pair Correlation Conjecture for the zeros of the Riemann zeta-function II: The Alternative Hypothesis. https://arxiv.org/abs/2507.06823

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