PreprintCC BY 4.0arXiv
The Riemann zeta function at the zeros of real-rooted trigonometric polynomials
Abstract
Let P be a trigonometric polynomial with only real zeros. We prove that at most 2/3 of its zeros coincide with the zeros of the Riemann zeta function ζ(s) on the critical line. We also show that supsubstackT≤ x≤2T P(x)=0 log |ζ(1/2+ix)| ≥ (1+o(1))√((log T)/(6loglog T)). These results extend the corresponding theorems of Li and Radziwill on the zeros of ζ(s) along vertical arithmetic progressions. The main novelty in our approach is the use of the work of Kurasov and Sarnak and Lee-Yang representations of Alon-Cohen-Vinzant, which show that the corresponding measure ∑P(γ)=0 bf δγ is crystalline.
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BibTeX
@misc{goncalves2026riemann,
title = {{The Riemann zeta function at the zeros of real-rooted trigonometric polynomials}},
author = {Felipe Gonçalves and Bryce Kerr and Oleksiy Klurman and Guilherme Vedana},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26705},
eprint = {2610.26705},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26705},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Riemann zeta function at the zeros of real-rooted trigonometric polynomials
AU - Felipe Gonçalves
AU - Bryce Kerr
AU - Oleksiy Klurman
AU - Guilherme Vedana
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26705
UR - https://arxiv.org/abs/2610.26705
AB - Let P be a trigonometric polynomial with only real zeros. We prove that at most 2/3 of its zeros coincide with the zeros of the Riemann zeta function ζ(s) on the critical line. We also show that sup_substackT≤ x≤2T P(x)=0 log |ζ(1/2+ix)| ≥ (1+o(1))√((log T)/(6loglog T)). These results extend the corresponding theorems of Li and Radziwill on the zeros of ζ(s) along vertical arithmetic progressions. The main novelty in our approach is the use of the work of Kurasov and Sarnak and Lee-Yang representations of Alon-Cohen-Vinzant, which show that the corresponding measure ∑_P(γ)=0 bf δ_γ is crystalline.
ER -