Constants of de Bruijn–Newman type in analytic number theory and statistical physics
Bulletin of the American Mathematical Society
One formulation in 1859 of the Riemann Hypothesis (RH) was that the Fourier transform H f (z) of f for z ∈ C has only real zeros when f (t) is a specific function Φ(t). Pólya's 1920s approach to the RH extended H f to H f,λ , the Fourier transform of e λt 2 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). Pólya's work together with 1950 and 1976 results of de Bruijn and Newman, respectively, imply the
... ively, imply the existence of a finite constant Λ DN = Λ DN (Φ) in (−∞, 1/2] such that H Φ,λ has only real zeros if and only if λ ≥ Λ DN ; the RH is then equivalent to Λ DN ≤ 0. Recent developments include the Rodgers and Tao proof of the 1976 conjecture that Λ DN ≥ 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 H ρ,λ and Λ DN (ρ) behaviors; some of these are new and based on a recent weak convergence theorem of the authors.