| 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 zeros of the general Euler-Riemann's ζ-function . In brief, the Riemann hypothesis is equivalent to the conjecture that Λ ≤ 0. De Bruijn showed in 1950 that ''H'' has only real zeros if λ ≥ 1/2, and moreover, that if ''H'' has only real zeros for some λ, ''H'' also has only real zeros if λ is replaced by any larger value. Newman proved in 1976 the existence of a constant Λ for which the "if and only if" claim holds; and this then implies that Λ is unique. Newman conjectured that Λ ≥ 0, an intriguing counterpart to the Riemann hypothesis. Serious calculations on lower bounds for Λ have been made since 1988 and—as can be seen from the table—are still being made: EXTERNAL LINKS |
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