An improved lower bound for the de Bruijn-Newman constant

Abstract

In this article, we report on computations that led to the discovery of a new Lehmer pair of zeros for the Riemann ζ \zeta function. Given this new close pair of zeros, we improve the known lower bound for de Bruijn-Newman constant Λ \Lambda . The Riemann hypothesis is equivalent to the assertion Λ ≤ 0 \Lambda \leq 0 . In this article, we establish that in fact we have Λ > − 1.14541 × 10 − 11 \Lambda > -1.14541 \times 10^{-11} . This new bound confirms the belief that if the Riemann hypothesis is true, it is barely true.