Jensen Polynomials for the Riemann Xi Function

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 $\mathrm{Re}(s)=\frac{1}{2}$. Pólya 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 $d\ge 1$, 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 $J^{d,n}(X)$.

13 pages. This revision represents a major revision of the previous version. The exposition has been improved and many clarifications have been added. Moreover, Theorem 1.1 has been improved

References in corpus (1)

• Zeros of Jensen polynomials and asymptotics for the Riemann xi function

Cited by in corpus (3)

• Zeros of Jensen polynomials and asymptotics for the Riemann xi function
• The Riemann hypothesis is true up to $3\cdot 10^{12}$
• Riemann Hypothesis, Modified Morse Potential and Supersymmetric Quantum Mechanics