Zeta Zeros on the Critical Line

2025/11/25 by D. A. Goldston, Goldston, Daniel A., Ade Irma Suriajaya +1
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2511.20059

Abstract

Montgomery in 1973 introduced the pair correlation method to study the vertical distribution of Riemann zeta-function zeros. This work assumed the Riemann Hypothesis (RH). One striking application was a short proof that at least 2/3 of zeta-zeros are simple zeros, the first result of its type. Over the last 50 years, most work on pair correlation of zeta-zeros has continued to assume RH. Here we show that if RH could be removed from Montgomery's simple zero proof, then this would also give a proof that 2/3 of the zeros are simple and on the critical line. This idea has been applied in several recent papers to obtain other results on the zeros.

Citations

• Pair Correlation Conjecture for the zeros of the Riemann zeta-function II: The Alternative Hypothesis
• 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
• Pair Correlation Estimates for the Zeros of the Zeta Function via\n Semidefinite Programming

Related

• Pair Correlation of Zeros of the Riemann Zeta Function I: Proportions of Simple Zeros and Critical Zeros
• Pair Correlation Conjecture for the Zeros of the Riemann Zeta-function I: Simple and Critical Zeros
• Pair Correlation Conjecture for the zeros of the Riemann zeta-function II: The Alternative Hypothesis
• Constructing a Proof of the Riemann Hypothesis
• Zeta zero dependence and the critical line