De Bruijn-Newman constant
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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:
| 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 |