An asymptotic Jensen-Polya criterion for GRH

Abstract

The Jensen--Pólya criterion expresses the generalized Riemann hypothesis (GRH) for Dedekind zeta-functions as a real-rootedness condition on their Jensen polynomials. Previous asymptotic results require large shifts, while computations establish real-rootedness in the Riemann case through a finite range of degrees. Here we study the unshifted family as the degree tends to infinity. For every number field $K$, we prove unconditionally that $J_{K,d}(X)$ is nonzero and has at least $d-C_K\sqrt{d\log d}$ distinct negative real zeros for all sufficiently large $d$, where $C_K>0$ depends only on $K$. Therefore, the proportion of real zeros tends to one with error $O_K(\sqrt{\log d/d})$. The main theorems have been autoformalized in Lean~4 by AxiomProver.