# De Bruijn–Newman constant

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The De Bruijn–Newman constant, denoted by Λ and named after Nicolaas Govert de Bruijn and Charles M. Newman, is a mathematical constant defined via the zeros of a certain function H(λz), where λ is a real parameter and z is a complex variable. 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. In brief, the Riemann hypothesis is equivalent to the conjecture that Λ ≤ 0.

De Bruijn showed in 1950 that H has only real zeros if λ ≥ 1/2, and moreover, that if H has only real zeros for some λ, H also has only real zeros if λ is replaced by any larger value. De Bruijn's upper bound of $\Lambda \leq 1/2$ was not improved until 2008, when Ki, Kim and Lee proved $\Lambda <1/2$ , making the inequality strict.

As of December 2018, the current best upper bound is $\Lambda \leq 0.22$ , achieved in the 15th Polymath project. A manuscript of the Polymath work was submitted to arXiv in late April 2019 , and has been accepted for publication by the journal Research In the Mathematical Sciences.

Newman proved in 1976 the existence of a constant Λ for which the "if and only if" claim holds; and this then implies that Λ is unique. Newman conjectured that Λ ≥ 0, an intriguing counterpart to the Riemann hypothesis. In January 2018 Brad Rodgers and Terence Tao published a paper on arXiv in which they claim that Λ ≥ 0. Thus if Riemann hypothesis is true, then the value of Λ must be 0.

Year Lower bound on Λ
1988 −50
1991 −5
1990 −0.385
1994 −4.379×10−6
1993 −5.895×10−9
2000 −2.7×10−9
2011 −1.1×10−11
2018 0

Since $H(\lambda ,z)$ is just the Fourier transform of $F(e^{\lambda x}\Phi )$ then H has the Wiener–Hopf representation:

$\xi (1/2+iz)=A{\sqrt {\pi }}(\lambda )^{-1}\int _{-\infty }^{\infty }e^{{\frac {-1}{4\lambda }}(x-z)^{2}}H(\lambda ,x)\,dx$ which is only valid for λ positive or 0, it can be seen that in the limit λ tends to zero then $H(0,x)=\xi (1/2+ix)$ for the case Lambda is negative then H is defined so:

$H(z,\lambda )=B{\sqrt {\pi }}(\lambda )^{-1}\int _{-\infty }^{\infty }e^{{\frac {-1}{4\lambda }}(x-z)^{2}}\xi (1/2+ix)\,dx$ where A and B are real constants.