De Bruijn-Newman constant

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:

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

$xi\; (1/2+iz)=\; Asqrt\; pi\; (lambda)^\{-1\}\; int\_\{-infty\}^\{infty\}dx\; e^\{frac\{-1\}\{4lambda\}(x-z)^\{2\; H(lambda\; ,\; x)$

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

$H(z,lambda)=Bsqrt\; pi\; (lambda)^\{-1\}\; int\_\{-infty\}^\{infty\}dx\; e^\{frac\{-1\}\{4lambda\}(x-z)^\{2\; xi(1/2+ix)$

Where A and B are real constant.

External links

*