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(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.[1] 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.[2] 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[3]
2000 −2.7×10−9[4]
2011 −1.1×10−12[5]

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

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

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

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

where A and B are real constants.

References

1. ^ de Bruijn, N.G. (1950). "The Roots of Triginometric Integrals". Duke Math. J. 17: 197–226. doi:10.1215/s0012-7094-50-01720-0. Zbl 0038.23302.
2. ^ Newman, C.M. (1976). "Fourier Transforms with only Real Zeros". Proc. Amer. Math. Soc. 61: 245–251. doi:10.1090/s0002-9939-1976-0434982-5. Zbl 0342.42007.
3. ^ Csordas, G.; Odlyzko, A.M.; Smith, W.; Varga, R.S. (1993). "A new Lehmer pair of zeros and a new lower bound for the De Bruijn–Newman constant Lambda" (pdf). Electronic Transactions on Numerical Analysis. 1: 104–111. Zbl 0807.11059. Retrieved June 1, 2012.
4. ^ Odlyzko, A.M. (2000). "An improved bound for the de Bruijn–Newman constant". Numerical Algorithms. 25: 293–303. Zbl 0967.11034.
5. ^ Saouter, Yannick; Gourdon, Xavier; Demichel, Patrick (2011). "An improved lower bound for the de Bruijn-Newman constant". Mathematics of Computation. 80 (276): 2281–2287. doi:10.1090/S0025-5718-2011-02472-5. MR 2813360.