On the de Bruijn-Newman constant

If λ⁽⁰⁾ denotes the infimum of the set of real numbers λ such that the entire function Ξ_λ represented byΞ_λ (t) = underover(∫, 0, ∞) e^{frac(λ, 4) (log x)2 + frac(i t, 2) log x} (x^{5 / 4} underover(∑, n = 1, ∞) (2 n⁴ π² x - 3 n² π) e^{- n2 π x}) frac(d x, x) has only real zeros, then the de Bruijn-Newman constant Λ is defined as Λ = 4 λ⁽⁰⁾. The Riemann hypothesis is equivalent to the inequality Λ ≤ 0. The fact that the non-trivial zeros of the Riemann zeta-function ζ lie in the strip {s : 0 < Re s < 1} and a theorem of de Bruijn imply that Λ ≤ 1 / 2. In this paper, we prove that all but a finite number of zeros of Ξ_λ are real and simple for each λ > 0, and consequently that Λ < 1 / 2.