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

Source: [Pair Correlation Conjecture for the zeros of the Riemann zeta-function II: The Alternative Hypothesis](<https://www.sota2.com/research/paper/68fa5dce1e209131ce9dcce5>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T18:25:11.159Z

Authors: Daniel A. Goldston, Junghun Lee, Jordan Schettler, Ade Irma Suriajaya

Published: 2025-07-09T13:12:19.000Z

## 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.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2507.06823>)

[PDF](<https://arxiv.org/pdf/2507.06823.pdf>)