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

One formulation in 1859 of the Riemann Hypothesis (RH) was that the Fourier transform Hf (z) of f for z ∈ C has only real zeros when f(t) is a specific function Φ(t). Polya's 1920s approach to the RH extended Hf to Hf, λ, the Fourier transform of eλt2 f(t). We review developments of this approach to the RH and related ones in statistical physics where f(t) is replaced by a measure dρ(t). Polya's work together with 1950 and 1976 results of de Bruijn and Newman, respectively, imply the existence of a finite constant ΛDN = ΛDN(Φ) in (-∞, 1/2] such that HΦ, λ has only real zeros if and only if λ ≥ ΛDN; the RH is then equivalent to ΛDN ≤ 0. Recent developments include the Rodgers and Tao proof of the 1976 conjecture that ΛDN ≥ 0 (that the RH, if true, is only barely so) and the Polymath 15 project improving the 1/2 upper bound to about 0.22. We also present examples of ρ's with differing Hρ, λ and ΛDN(ρ) behaviors; some of these are new and based on a recent weak convergence theorem of the authors.