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's hypothesis concerning the zeros of the Riemann zeta-function: since the Riemann hypothesis is equivalent to the claim that all the zeroes of ${\displaystyle H(0,z)}$ are real, the Riemann hypothesis is equivalent to the conjecture that ${\displaystyle \Lambda \leq 0}$.^[1] Brad Rodgers and Terence Tao proved that ${\displaystyle \Lambda \geq 0}$, so Riemann's hypothesis is equivalent to ${\displaystyle \Lambda =0}$.^[2] A simplified proof of the Rodgers–Tao result was later given by Alexander Dobner.^[3]

De Bruijn showed in 1950 that ${\displaystyle H}$ has only real zeros if ${\displaystyle \lambda \geq 1/2}$, and moreover, that if ${\displaystyle H}$ has only real zeros for some ${\displaystyle \lambda }$, ${\displaystyle H}$ also has only real zeros if ${\displaystyle \lambda }$ is replaced by any larger value.^[4] Newman proved in 1976 the existence of a constant ${\displaystyle \Lambda }$ for which the "if and only if" claim holds; and this then implies that ${\displaystyle \Lambda }$ is unique. Newman also conjectured that ${\displaystyle \Lambda \geq 0}$,^[5] which was then proven by Brad Rodgers and Terence Tao in 2018.

De Bruijn's upper bound of ${\displaystyle \Lambda \leq 1/2}$ was not improved until 2008, when Ki, Kim and Lee proved ${\displaystyle \Lambda <1/2}$, making the inequality strict.^[6]

In December 2018, the 15th Polymath project improved the bound to ${\displaystyle \Lambda \leq 0.22}$.^[7][8][9] A manuscript of the Polymath work was submitted to arXiv in late April 2019,^[10] and was published in the journal Research In the Mathematical Sciences in August 2019.^[11]

This bound was further slightly improved in April 2020 by Platt and Trudgian to ${\displaystyle \Lambda \leq 0.2}$.^[12]