The De Bruijn-Newman constant is non-negative

2018/01/18 by Brad Rodgers, Terence Tao, Rodgers, Brad +1 · 3 citations
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Physics Problems #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1801.05914

Abstract

For each t ∈ \bf R, define the entire function H_t(x) := ∫₀^∞ e^tu2 Φ(u) cos(xu) du where Φ is the super-exponentially decaying function Φ(u) := ∑_n=1^∞ (2π² n⁴ e^9u - 3πn² e^5u ) exp(-πn² 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 H₀ 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.

Cited by

• Zeros of polynomial powers under the heat flow
• Zero distribution of multiplicative Hermite and Laguerre polynomials
• Lehmer pairs and binomial series

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