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 connected with 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:

ower bound on Λ
a href="/encyclopedia/1988" title="1988">1988	-50
a href="/encyclopedia/1991" title="1991">1991	-5
a href="/encyclopedia/1990" title="1990">1990	-0.385
a href="/encyclopedia/1994" title="1994">1994	-4.379 · 10 ^-6
a href="/encyclopedia/1993" title="1993">1993	-5.895 · 10 ^-9
a href="/encyclopedia/2000" title="2000">2000	-2.7 · 10 ^-9

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