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

2019
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Bulletin of the American Mathematical Society
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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

doi:10.1090/bull/1668
fatcat:372uk5p7nbgzbatwl4l6lvzqjm