Jensen polynomials for the Riemann zeta function and other sequences

The Pólya-Jensen criterion for the Riemann hypothesis asserts that R H is equivalent to the hyperbolicity of certain Jensen polynomials for all degrees d ≥1 and all shifts n . For each degree d ≥1 , we confirm this criterion for all sufficiently large shifts n . This represents a theoretical advance in the field. The method of proof is rooted in the newly discovered phenomenon that these polynomials are nicely approximated by Hermite polynomials. Furthermore, it is shown that this method applies to a large class of related problems.