All but finitely many of the fixed degree Jensen polynomials for the Riemann zeta function are hyperbolic

You need to know: Set ${\mathbb C}$ of complex numbers, real part $\text{Re}(z)$ of complex number z, function of complex variable, infinite series, integration, meromorphic function, analytic continuation, notation ${{d}\choose{j}}=\frac{d!}{j!(n-j)!}$ .

Background: For $z \in{\mathbb C}$ with $\text{Re}(z)>1$ , let $\zeta(z)=\sum\limits_{n=1}^\infty n^{-z}$ . By analytic continuation, function $\zeta(z)$ can be extended to a meromorphic function on the whole ${\mathbb C}$ , and it is called the Riemann zeta function. Similarly, let $\Gamma(z)$ be the analytic continuation of integral $\Gamma(z)=\int\limits_0^\infty x^{z-1}e^{-x}dx$ , defined for $\text{Re}(z)>0$ . Let $\Lambda(z)=\pi^{-z/2}\Gamma(z/2)\zeta(z)$ , and let sequence $\{\gamma(n)\}_{n=0}^\infty$ be defined by $(-1+4z^2)\Lambda\left(\frac{1}{2}+z\right)=\sum\limits_{n=0}^\infty\frac{\gamma(n)}{n!}z^{2n}$ . The Jensen polynomial for the Riemann zeta function of degree d and shift n is the polynomial $J_{\gamma}^{d,n}(x)=\sum\limits_{j=0}^n {{d}\choose{j}}\gamma(n+j)x^j$ . We say that a polynomial with real coefficients is hyperbolic if all of its zeros are real.

The Theorem: On 12th February 2019, Michael Griffin, Ken Ono, Larry Rolen, and Don Zagier submitted to the Proceedings of the National Academy of Sciences a paper in which they proved that for any $d\geq 1$ there is a constant $N=N(d)$ , such that the polynomial $J_{\gamma}^{d,n}(x)$ is hyperbolic for all $n\geq N$ .

Short context: The Riemann hypothesis (RH) states that if $\zeta(z)=0$ then either $z=-2k$ for some integer $k>0$ or $\text{Re}(z)=\frac{1}{2}$ . It is one of the most important open problems in the whole mathematics, and has many equivalent formulations. One of the equivalent formulations of RH, established by Pólya in 1927, states that polynomials $J_{\gamma}^{d,n}(x)$ are hyperbolic for all integers $d\geq 0$ and $n\geq 0$ . Before 2019, this statement was known to hold only for $d\leq 3$ . The Theorem proves the hyperbolicity of $J_{\gamma}^{d,n}(x)$ for all d, assuming that n is sufficiently large (depending on d). As a corollary, the authors also proved this for all n if $d\leq 8$ .