TY - JOUR

T1 - Constants of de Bruijn-Newman type in analytic number theory and statistical physics

AU - Newman, Charles M.

AU - Wu, Wei

N1 - Funding Information:
Received by the editors March 3, 2019. 2010 Mathematics Subject Classification. Primary 11M26, 30C15, 60K35. The research reported here was supported in part by US NSF grant DMS-1507019.
Funding Information:
We thank Ivan Corwin for the invitation to submit this paper to the Bulletin of the AMS and for comments on an earlier draft. We also thank Louis-Pierre Arguin and two anonymous reviewers for their detailed c
Publisher Copyright:
© 2019 American Mathematical Society.

PY - 2020/10

Y1 - 2020/10

N2 - 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.

AB - 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.

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U2 - 10.1090/BULL/1668

DO - 10.1090/BULL/1668

M3 - Article

AN - SCOPUS:85077694312

VL - 57

SP - 595

EP - 614

JO - Bulletin of the American Mathematical Society

JF - Bulletin of the American Mathematical Society

SN - 0273-0979

IS - 4

ER -