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The de Bruijn–Newman constant, denoted by Λ and named after Nicolaas Govert de Bruijn and Charles M. Newman, is a mathematical constant defined via the zeros of a certain function H, where λ is a real parameter and z is a complex variable. More precisely,
where Φ {\displaystyle \Phi } is the super-exponentially decaying function
and Λ is the unique real number with the property that H has only real zeros if and only if λ ≥ Λ.
The constant is closely connected with Riemann's hypothesis concerning the zeros of the Riemann zeta-function: since the Riemann hypothesis is equivalent to the claim that all the zeroes of H are real, the Riemann hypothesis is equivalent to the conjecture that Λ ≤ 0. Brad Rodgers and Terence Tao proved that Λ < 0 cannot be true, so Riemann's hypothesis is equivalent to Λ = 0. A simplified proof of the Rodgers–Tao result was later given by Alexander Dobner.