Consider the de-Bruijn-Newman function

$$H_{t}(z)=\int_{0}^{\infty}e^{tu^2}\phi(u)\cos (zu) \mathrm{du},$$ where $\phi$ is a certain super exponentially decaying function. Much work has been done on the function $H$, including in the Polymath15 project.

My question: Let $z=x+iy$. Is there a known $y_0$ such that $H_{t}(x+iy)\neq 0$ for ALL $t<0$ and $|y|\geq y_0$ ?

  • $\begingroup$ The paper for Polymath15, Effective approximation of heat flow evolution of the Riemann ξ function, and a new upper bound for the de Bruijn-Newman constant is finished and submitted for publication, and available on the arXiv as arxiv.org/abs/1904.12438 $\endgroup$ – David Roberts Jun 5 at 4:42

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