# On the complex zeros of the De-Bruijn-Newman function

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 $$|y|\geq y_0$$ ?

• 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 – David Roberts Jun 5 at 4:42