Jensen polynomials are not a plausible route to proving the Riemann Hypothesis

TL;DR

This paper critically evaluates recent claims linking Jensen polynomials to the Riemann Hypothesis, concluding that these polynomials are unlikely to be useful for proving the conjecture and clarifying the limitations of such approaches.

Contribution

It clarifies the limitations of Jensen polynomials in relation to the Riemann Hypothesis and proposes criteria to assess the usefulness of polynomial-based equivalences.

Findings

No justification for Jensen polynomials proving the Riemann Hypothesis

Hermite polynomial appearance may indicate a new universal law

Jensen polynomials are unlikely to be effective for attacking the Riemann Hypothesis

Abstract

Recent work on the Jensen polynomials of the Riemann xi-function and its derivatives found a connection to the Hermite polynomials. Those results have been suggested to give evidence for the Riemann Hypothesis, and furthermore it has been suggested that those results shed light on the random matrix statistics for zeros of the zeta-function. We place that work in the context of prior results, and explain why the appearance of Hermite polynomials is interesting and surprising, and may represent a new type of universal law which refines M. Berry's "cosine is a universal attractor" principle. However, we find there is no justification for the suggested connection to the Riemann Hypothesis, nor for the suggested connection to the conjectured random matrix statistics for zeros of L-functions. These considerations suggest that Jensen polynomials, as well as a large class of related polynomials, are not useful for attacking the Riemann Hypothesis. We propose general criteria for determining whether an equivalence to the Riemann Hypothesis is likely to be useful.