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(eλxΦ) 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 constant.
