de Bruijn–Newman constant

The de Bruijn–Newman constant, denoted by Λ and named after Nicolaas Govert de Bruijn and Charles Michael Newman, is a mathematical constant defined via the zeros of a certain function H(λ,z), where λ is a real parameter and z is a complex variable. More precisely,

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

where ${\displaystyle \Phi }$ is the super-exponentially decaying function

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

and Λ is the unique real number with the property that H has only real zeros if and only if λ≥Λ.

The constant is closely connected with Riemann's hypothesis concerning the zeros of the Riemann zeta-function: since the Riemann hypothesis is equivalent to the claim that all the zeroes of H(0, z) are real, the Riemann hypothesis is equivalent to the conjecture that Λ≤0. Brad Rodgers and Terence Tao proved that Λ<0 cannot be true, so Riemann's hypothesis is equivalent to Λ = 0. A simplified proof of the Rodgers–Tao result was later given by Alexander Dobner.