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

Since H(λ,z) is just the Fourier transform of F(e^λxΦ) 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) = ξ(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.

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, 293-303, 2000

External links

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