De Bruijn-Newman constant

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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. 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(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 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

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