# An unconditional Montgomery Theorem for Pair Correlation of Zeros of the Riemann Zeta Function

Source: [An unconditional Montgomery Theorem for Pair Correlation of Zeros of the Riemann Zeta Function](<https://www.sota2.com/research/paper/68fa5c191e209131ce9ce238>)

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

Data updated: 2025-10-25T17:26:27.186Z

Authors: Siegfred Alan C. Baluyot, Daniel Alan Goldston, Ade Irma Suriajaya, Caroline L. Turnage-Butterbaugh

Published: 2023-06-07T21:47:10.000Z

## Abstract

Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem concerning pair correlation of zeros of the Riemann zeta-function. One consequence of this theorem is that, assuming RH, at least $67.9\\%$ of the nontrivial zeros are simple. Here we obtain an unconditional form of Montgomery's theorem and show how to apply it to prove the following result on simple zeros: Assuming all the zeros $\\rho=\\beta+i\\gamma$ of the Riemann zeta-function such that $T^{3/8}\<\\gamma\\le T$ satisfy $\|\\beta-1/2\|\<1/(2\\log T)$, %lie in the thin box $\\{s=\\sigma +it: \|\\sigma-1/2\|\<1/(2\\log T),\\ T^{3/8}\<t\\le T\\}$, then, as $T$ tends to infinity, at least $61.7\\%$ of these zeros are simple. The method of proof neither requires nor provides any information on whether any of these zeros are on or not on the critical line where $\\beta=1/2$. We also obtain the same result under the weaker assumption of a strong zero-density hypothesis.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2306.04799>)

[PDF](<https://arxiv.org/pdf/2306.04799.pdf>)