De Bruijn–Newman constant

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 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
2011	−1.1×10−12

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.

External links

• Weisstein, Eric W. "de Bruijn–Newman Constant". MathWorld.