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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 conected with the Riemann's hypothesis on the zeroes of the general Euler - Riemann's ζ-function. In brief, the Riemann hypothesis is equivalent to the conjecture that Λ ≤ 0.

De Bruijn in 1950 showed that Λ ≤ 1/2, according to Newman's work, who first estimated it would be Λ ≥ 0. Serious calculations on Λ have been made since 1988 and are still being made as we see from the table:

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

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