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


Authors: Charles M. Newman and Wei Wu
Journal: Bull. Amer. Math. Soc.
MSC (2010): Primary 11M26, 30C15, 60K35
DOI: https://doi.org/10.1090/bull/1668
Published electronically: April 19, 2019
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Abstract: One formulation in 1859 of the Riemann Hypothesis (RH) was that the Fourier transform $ H_f(z)$ of $ f$ for $ z \in \mathbb{C}$ has only real zeros when $ f(t)$ is a specific function $ \Phi (t)$. Pólya's 1920s approach to the RH extended $ H_f$ to $ H_{f,\lambda }$, the Fourier transform of $ e^{\lambda t^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 $ d \rho (t)$. Pólya's work together with 1950 and 1976 results of de Bruijn and Newman, respectively, imply the existence of a finite constant $ \Lambda _{DN} = \Lambda _{DN} (\Phi )$ in $ (-\infty , 1/2]$ such that $ H_{\Phi ,\lambda }$ has only real zeros if and only if $ \lambda \geq \Lambda _{DN}$; the RH is then equivalent to $ \Lambda _{DN} \leq 0$. Recent developments include the Rodgers and Tao proof of the 1976 conjecture that $ \Lambda _{DN} \geq 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 $ \rho $'s with differing $ H_{\rho ,\lambda }$ and $ \Lambda _{DN} (\rho )$ behaviors; some of these are new and based on a recent weak convergence theorem of the authors.


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Additional Information

Charles M. Newman
Affiliation: Courant Institute, New York University; and NYU Shanghai, People’s Republic of China

Wei Wu
Affiliation: University of Warwick, Warwick, United Kingdom

DOI: https://doi.org/10.1090/bull/1668
Received by editor(s): March 3, 2019
Published electronically: April 19, 2019
Additional Notes: The research reported here was supported in part by US NSF grant DMS-1507019.
Article copyright: © Copyright 2019 American Mathematical Society