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 $({\lambda }_n{)}_{n\ge 0}$. In this paper, we prove that the sequence $({\lambda }_n{)}_{n\ge 0}$ 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.

arXiv comment60 pages