On the de Bruijn–Newman constant

If λ(0)λ(0) denotes the infimum of the set of real numbers λ   such that the entire function ΞλΞλ represented byΞλ(t)=∫0∞eλ4(logx)2+it2logx(x5/4∑n=1∞(2n4π2x−3n2π)e−n2πx)dxx has only real zeros, then the de Bruijn–Newman constant Λ   is defined as Λ=4λ(0)Λ=4λ(0). The Riemann hypothesis is equivalent to the inequality Λ⩽0Λ⩽0. The fact that the non-trivial zeros of the Riemann zeta-function ζ   lie in the strip {s:00λ>0, and consequently that Λ<1/2Λ<1/2.