de Bruijn–Newman constant
Mathematical constant / From Wikipedia, the free encyclopedia
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The de Bruijn–Newman constant, denoted by and named after Nicolaas Govert de Bruijn and Charles Michael Newman, is a mathematical constant defined via the zeros of a certain function , where is a real parameter and is a complex variable. More precisely,
- ,
where is the super-exponentially decaying function
and is the unique real number with the property that has only real zeros if and only if .
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 are real, the Riemann hypothesis is equivalent to the conjecture that .[1] Brad Rodgers and Terence Tao proved that , so Riemann's hypothesis is equivalent to .[2] A simplified proof of the Rodgers–Tao result was later given by Alexander Dobner.[3]