The De Bruijn–newman Constant Is Non-Negative

Abstract

For each t ∈ R, define the entire functionwhere Φ is the super-exponentially decaying function(2π 2 n 4 e 9u − 3πn 2 e 5u ) exp(−πn 2 e 4u ).Newman showed that there exists a finite constant Λ (the de Bruijn-Newman constant) such that the zeroes of H t are all real precisely when t ≥ Λ. The Riemann hypothesis is the equivalent to the assertion Λ ≤ 0, and Newman conjectured the complementary bound Λ ≥ 0.In this paper we establish Newman's conjecture. The argument proceeds by assuming for contradiction that Λ < 0, and then analyzing the dynamics of zeroes of H t (building on the work of Csordas, Smith, and Varga) to obtain increasingly strong control on the zeroes of H t in the range Λ < t ≤ 0, until one establishes that the zeroes of H 0 are in local equilibrium, in the sense that locally behave (on average) as if they were equally spaced in an arithmetic progression, with gaps staying close to the global average gap size. But this latter claim is inconsistent with the known results about the local distribution of zeroes of the Riemann zeta function, such as the pair correlation estimates of Montgomery.

Constants of de Bruijn–Newman type in analytic number theory and statistical physics

“…More recently, Rodgers and Tao [RoT18] proved that Λ DN ≥ 0, thus confirming the Newman's conjecture. Instead of looking at individual pairs of zeros, they focused on zeros in intervals of the type [T, T + α], where 1 ≪ α ≪ log T and proved that they exhibit a kind of relaxation to local equilibrium -if Λ DN < 0, then the zeros of H 0 would be arranged locally as an approximate arithmetic progression.…”

“…A one-parameter family of solutions to (22), that are called the equilibrium state, is given by H t (z) = e tu 2 cos zu, for u > 0, whose zeros are all arranged as an arithmetic progression 2π(k+ 1 2 ) u : k ∈ Z . As discussed in Section 4 of [RoT18], (23) is reminiscent of Dyson Brownian motion and of similar behavior that was studied by Erdos, Schlein and Yau, in the context of gradient flow for the eigenvalues of random matrices [ESY11]. We now sketch some of the main steps in [RoT18].…”

“…Taking f to be a specific function Φ (see (7) below), RH is equivalent to all the zeros of H Φ,0 being real -i.e., equivalent to P Φ ∋ 0. In Section 2 we review, following the history from 1859 [Rie59] to the present [RoT18], some of the main developments of this approach to RH, including (see Subsections 2.5-2.6) those with motivations arising from statistical physics. Three key facts are 1. λ ∈ P Φ and λ ′ > λ imply λ ′ ∈ P Φ ; this is due to Pólya [P27] -see Theorem 2.…”

“…In Subsection 2.6 we survey the history, starting with [CNV88], of improvements on the [Ne76] lower bound Λ DN > −∞ culminating with the 2018 proof by Rodgers and Tao [RoT18] of the 1976 conjecture that Λ DN ≥ 0. Then in Subsection 2.7 we discuss the shorter history of improvements to the [DB50] upper bound Λ DN ≤ 1/2, consisting of the Ki-Kim-Lee [KKL09] bound Λ DN < 1/2 (and related results such as Theorem 14 below) as well as current efforts (see [T18]) by Tao and collaborators to obtain a concrete upper bound in (0, 1/2), expected at the time of this writing to be about 0.22.…”

Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman

“…Computeraided rigorous calculations of a lower bound for the De Bruijn-Newman constant have been made over the years. Until 2018, the best lower bound was −1.1 × 10 −11 [251], but recently Brad Rodgers and Terry Tao [247] posted a proof verifying Chuck's conjecture that Λ ≥ 0.…”