  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 $\Phi (t)$. P\'{o}lya's 1920s approach to 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 RH and related ones in
statistical physics where $f(t)$ is replaced by a measure $d \rho (t)$.
P\'{o}lya'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 $\lambda \geq \Lambda_{DN}$; 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 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.
