# Effective approximation of heat flow evolution of the Riemann $$\xi$$ function, and a new upper bound for the de Bruijn–Newman constant

@article{Polymath2019EffectiveAO,
title={Effective approximation of heat flow evolution of the Riemann \$\$\xi \$\$ function, and a new upper bound for the de Bruijn–Newman constant},
author={D. H. J. Polymath},
journal={Research in the Mathematical Sciences},
year={2019}
}
• D. Polymath
• Published 29 April 2019
• Mathematics
• Research in the Mathematical Sciences
For each $t \in \mathbf{R}$, define the entire function $$H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du$$ where $\Phi$ is the super-exponentially decaying function $$\Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} - 3\pi n^2 e^{5u} ) \exp(-\pi n^2 e^{4u} ).$$ This is essentially the heat flow evolution of the Riemann $\xi$ function. From the work of de Bruijn and Newman, there exists a finite constant $\Lambda$ (the \emph{de Bruijn-Newman constant}) such that the zeroes of $H_t$ are…
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