# Jensen polynomials for the Riemann zeta function and other sequences

## TL;DR

This paper proves the hyperbolicity of Jensen polynomials associated with the Riemann zeta function for degrees up to 8, using asymptotic formulas and Hermite polynomial modeling, supporting the Riemann Hypothesis.

## Contribution

It introduces a general theorem modeling Jensen polynomials with Hermite polynomials, proving hyperbolicity for degrees up to 8 and confirming related conjectures.

## Key findings

- Hyperbolicity proven for degrees up to 8
- Asymptotic formula for zeta derivatives accurate to all orders
- Supports GUE random matrix model prediction

## Abstract

In 1927 P\'olya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function $\zeta(s)$ at its point of symmetry. This hyperbolicity has been proved for degrees $d\leq 3$. We obtain an asymptotic formula for the central derivatives $\zeta^{(2n)}(1/2)$ that is accurate to all orders, which allows us to prove the hyperbolicity of a density $1$ subset of the Jensen polynomials of each degree. Moreover, we establish hyperbolicity for all $d\leq 8$. These results follow from a general theorem which models such polynomials by Hermite polynomials. In the case of the Riemann zeta function, this proves the GUE random matrix model prediction in derivative aspect. The general theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function.