De Bruijn-Newman constant

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The De Bruijn-Newman constant, denoted by Λ, is a mathematical constant and is defined via the zeros of a certain function H(λ, z), 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 on the zeros of the general Euler-Riemann's ζ-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. 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. Serious calculations on lower bounds for Λ have been made since 1988 and—as can be seen from the table—are still being made:

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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

Since $H (λ, z)$ is just the Fourier transform of $F (e λ x Φ)$ then H has the Wiener-Hopf representation:

$\xi (1/2+iz)= A\sqrt \pi (\lambda)^{-1} \int_{-\infty}^{\infty}dx e^{\frac{-1}{4\lambda}(x-z)^{2}} H(\lambda , x)$

which is only valid for lambda positive or 0, it can be seen that in the limit lambda tends to zero then $H (0, x) = ξ(1 / 2 + i x)$ for the case Lambda is negative then H is defined so:

$H(z,\lambda)=B\sqrt \pi (\lambda)^{-1} \int_{-\infty}^{\infty}dx e^{\frac{-1}{4\lambda}(x-z)^{2}} \xi(1/2+ix)$