Jensen Polynomials for the Riemann Xi Function

We investigate Riemann's xi function $ξ(s):=\frac{1}{2}s(s-1)π^{-\frac{s}{2}}Γ(\frac{s}{2})ζ(s)$ (here $ζ(s)$ is the Riemann zeta function). The Riemann Hypothesis (RH) asserts that if $ξ(s)=0$, then $\mathrm{Re}(s)=\frac{1}{2}$. Pólya proved that RH is equivalent to the hyperbolicity of the Jensen polynomials $J^{d,n}(X)$ constructed from certain Taylor coefficients of $ξ(s)$. For each $d\geq 1$, recent work proves that $J^{d,n}(X)$ is hyperbolic for sufficiently large $n$. Here we make this result effective. Moreover, we show how the low-lying zeros of the derivatives $ξ^{(n)}(s)$ influence the hyperbolicity of $J^{d,n}(X)$.

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