De Bruijn–Newman constant

The de Bruijn–Newman constant, denoted by ${\displaystyle \Lambda }$ and named after Nicolaas Govert de Bruijn and Charles Michael Newman, is a mathematical constant defined via the zeros of a certain function ${\displaystyle H(\lambda ,z)}$, where ${\displaystyle \lambda }$ is a real parameter and ${\displaystyle z}$ 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 ${\displaystyle \Lambda }$ is the unique real number with the property that ${\displaystyle H}$ has only real zeros if and only if ${\displaystyle \lambda \geq \Lambda }$.

The constant is closely connected with Riemann hypothesis. Indeed, the Riemann hypothesis is equivalent to the conjecture that ${\displaystyle \Lambda \leq 0}$. Brad Rodgers and Terence Tao proved that ${\displaystyle \Lambda \geq 0}$, so the Riemann hypothesis is equivalent to ${\displaystyle \Lambda =0}$. A simplified proof of the Rodgers–Tao result was later given by Alexander Dobner.