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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