de Bruijn–Newman constant

Description of constant

Define

$H(\lambda, z):=\int_{0}^{\infty} e^{\lambda u^{2}} \Phi(u) \cos (z u) \, du$,

where $\Phi$ is the super-exponential function decaying function

$\Phi(u) = \sum_{n=1}^{\infty} (2\pi^2n^4e^{9u}-3\pi n^2 e^{5u} ) e^{-\pi n^2 e^{4u}}.$

Newman showed in [N1976] that there exists a finite constant $C_{21}$ (the de Bruijn–Newman constant) such that the zeros of $H$ are all real precisely when $\lambda \geq C_{21}$.

Known upper bounds

Bound Reference Comments
0.5   [B1950]
< 0.5 [KKL2009]  
0.22 [P2019] Polymath project
0.2 [PT2021]  

Known lower bounds

Bound Reference Comments
$-50$ [CNV1987]  
$-5$ [RH1990]  
$-0.0991$ [CRV1991]  
$-5.895 \cdot 10 ^ {-9}$ [COSV1993]  
$-2.7 \cdot 10 ^ {-9}$ [O2000]  
$-1.1 \cdot 10 ^{-11}$ [SGD2011]  
0 [RT2020] Simplified in [D2020]

Additional comments

References