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Infinite log-concavity of the Taylor coefficients of the Riemann xi-function
Abstract
The Riemann hypothesis is equivalent to F(x) belonging to the Laguerre--Pólya class. Brändén [J. Reine Angew. Math., 2011] proved that if an entire function in the Laguerre--Pólya class has nonnegative Taylor coefficients, then its coefficient sequence is infinitely log-concave. Consequently, the Riemann hypothesis implies the infinite log-concavity of (λn)n≥0. In this paper, we prove that the sequence (λn)n≥0 is strictly infinitely log-concave. This resolves a conjecture of Zhu [Math. Z., 2023]. The proof combines explicit complex-analytic estimates for the iterated logarithmic ratios, rigorous interval arithmetic for a finite range of indices, and a global closure argument.
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BibTeX
@misc{liu2026infinite,
title = {{Infinite log-concavity of the Taylor coefficients of the Riemann xi-function}},
author = {Yanxin Liu and Jianxi Mao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04972},
eprint = {2610.04972},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04972},
}
RIS
TY - UNPB
TI - Infinite log-concavity of the Taylor coefficients of the Riemann xi-function
AU - Yanxin Liu
AU - Jianxi Mao
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.04972
UR - https://arxiv.org/abs/2610.04972
AB - The Riemann hypothesis is equivalent to F(x) belonging to the Laguerre--Pólya class. Brändén [J. Reine Angew. Math., 2011] proved that if an entire function in the Laguerre--Pólya class has nonnegative Taylor coefficients, then its coefficient sequence is infinitely log-concave. Consequently, the Riemann hypothesis implies the infinite log-concavity of (λ_n)_n≥0. In this paper, we prove that the sequence (λ_n)_n≥0 is strictly infinitely log-concave. This resolves a conjecture of Zhu [Math. Z., 2023]. The proof combines explicit complex-analytic estimates for the iterated logarithmic ratios, rigorous interval arithmetic for a finite range of indices, and a global closure argument.
ER -