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Jensen Polynomials for the Riemann Xi Function (1910.01227v3)
Published 2 Oct 2019 in math.NT
Abstract: We investigate Riemann's xi function $\xi(s):=\frac{1}{2}s(s-1)\pi{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)$ (here $\zeta(s)$ is the Riemann zeta function). The Riemann Hypothesis (RH) asserts that if $\xi(s)=0$, then $\mathrm{Re}(s)=\frac{1}{2}$. P\'olya proved that RH is equivalent to the hyperbolicity of the Jensen polynomials $J{d,n}(X)$ constructed from certain Taylor coefficients of $\xi(s)$. 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 $\xi{(n)}(s)$ influence the hyperbolicity of $J{d,n}(X)$.