An unconditional Montgomery theorem for pair correlation of zeros of the Riemann zeta-function
Publication Date
1-1-2024
Document Type
Article
Publication Title
Acta Arithmetica
Volume
214
DOI
10.4064/aa230612-20-3
First Page
357
Last Page
376
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: If all the zeros ρ = β + iγ of the Riemann zeta-function such that T3/8 < γ ≤ T satisfy |β − 1/2| < 1/(2 log 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 or are not on the critical line where β = 1/2. We also obtain the same result under the weaker assumption of a strong zero-density hypothesis.
Funding Number
DMS-1854398
Funding Sponsor
American Institute of Mathematics
Keywords
pair correlation, Riemann zeta-function, simple zeros, zero-density, zeros
Department
Mathematics and Statistics
Recommended Citation
Siegfred Alan C. Baluyot, Daniel Alan Goldston, Ade Irma Suriajaya, and Caroline L. Turnage-Butterbaugh. "An unconditional Montgomery theorem for pair correlation of zeros of the Riemann zeta-function" Acta Arithmetica (2024): 357-376. https://doi.org/10.4064/aa230612-20-3
