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 $Φ(t)$. Pólya's 1920s approach to RH extended $H_f$ to $H_{f,λ}$, the Fourier transform of $e^{λt^2} f(t)$. We review developments of this approach to RH and related ones in statistical physics where $f(t)$ is replaced by a measure $d ρ(t)$. Pólya's work together with 1950 and 1976 results of de Bruijn and Newman, respectively, imply the existence of a finite constant $Λ_{DN} = Λ_{DN} (Φ)$ in $(-\infty, 1/2]$ such that $H_{Φ,λ}$ has only real zeros if and only if $λ\geq Λ_{DN}$; RH is then equivalent to $Λ_{DN} \leq 0$. Recent developments include the Rodgers and Tao proof of the 1976 conjecture that $Λ_{DN} \geq 0$ (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 $ρ$'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.