Jensen polynomials for the Riemann xi-function

We investigate ξ(s)=12s(s−1)π−s2Γ(s2)ζ(s), where ζ(s) is the Riemann zeta function. The Riemann hypothesis (RH) asserts that if ξ(s)=0, then Re(s)=12. Pólya proved that RH is equivalent to the hyperbolicity of the Jensen polynomials Jd,n(X) constructed from certain Taylor coefficients of ξ(s). For each d≥1, recent work proves that Jd,n(X) is hyperbolic for sufficiently large n. In this paper, we make this result effective. Moreover, we show how the low-lying zeros of the derivatives ξ(n)(s) influence the hyperbolicity of Jd,n(X).