De BruijnNewman constant
- The correct title of this article is de Bruijn-Newman constant. The initial letter is capitalized due to technical restrictions.
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 can be seen 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} |
External links
Mathworld article on the de Bruijn-Newman Constant
fr:Constante de De Bruijn-Newman it:Costante Bruijn-Newman pl:Stała de Bruijna-Newmana