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. 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:

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(\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 lambda positive or 0, it can be seen that in the limit lambda 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.


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