On the de Bruijn–Newman constant

Abstract

If λ ( 0 ) denotes the infimum of the set of real numbers λ such that the entire function Ξ λ represented by Ξ λ ( t ) = ∫ 0 ∞ e λ 4 ( log x ) 2 + i t 2 log x ( x 5 / 4 ∑ n = 1 ∞ ( 2 n 4 π 2 x − 3 n 2 π ) e − n 2 π x ) 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 .