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(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 concerning the zeros of the Riemann zetafunction. 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:
Since H(λ,z) is just the Fourier transform of F(e^{λx}Φ) then H has the Wiener–Hopf representation:
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 + ix) for the case Lambda is negative then H is defined so:
where A and B are real constants.
References
 Csordas & Odlyzko & Smith & Varga, A new Lehmer pair of zeros and a new lower bound for the De Bruijn–Newman constant Lambda, Electronic Transactions on Numerical Analysis, T1, p104–111, 1993
 N.G. de Bruijn, The Roots of Triginometric Integrals, Duke Math. J. 17, 197–226, 1950
 C.M. Newman, Fourier Transforms with only Real Zeros, Proc. Amer. Math. Soc. 61, 245–251, 1976
 A.M. Odlyzko, An improved bound for the de Bruijn–Newman constant, Numerical Algorithms 25, 293303, 2000
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