On the de Bruijn-Newman constant

If $\lambda^{(0)}$ denotes the infimum of the set of real numbers $\lambda$ such that the entire function $\Xi_{\lambda}$ represented by
$$ \Xi_{\lambda}(t) = \int_{0}^{\infty} e^{\frac{\lambda}{4}(\log x)^2 + \frac{it}{2}\log x}\left( x^{5/4}\sum_{n=1}^{\infty}\left(2n^4 \pi^2 x - 3n^2\pi\right)e^{-n^2 \pi x}\right)\frac{dx}{x} $$
has only real zeros, then the de Bruijn-Newman constant $\Lambda$ is defined as $\Lambda=4\lambda^{(0)}$. The Riemann hypothesis is equivalent to the inequality $\Lambda\leqslant 0$. Recently, Rodgers and Tao proved $\Lambda\geqslant 0$. In this talk, concerning this, I will introduce results with Young-One Kim and Jungseob Lee.

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