Infinite log-concavity of the Taylor coefficients of the Riemann xi-function

Yanxin Liu; Jianxi Mao

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arXiv · 2610.04972

Infinite log-concavity of the Taylor coefficients of the Riemann xi-function

Yanxin Liu·Jianxi Mao

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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 $(λ_n)_{n\ge0}$. In this paper, we prove that the sequence $(λ_n)_{n\ge0}$ 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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Yanxin Liu, Jianxi Mao. 2026-10-04. Infinite log-concavity of the Taylor coefficients of the Riemann xi-function. https://arxiv.org/abs/2610.04972

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