Jensen polynomials are not a plausible route to proving the Riemann Hypothesis
Abstract
Recent work on the Jensen polynomials of the Riemann xifunction 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 zetafunction. 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 Lfunctions. 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2020
 DOI:
 10.48550/arXiv.2008.07206
 arXiv:
 arXiv:2008.07206
 Bibcode:
 2020arXiv200807206F
 Keywords:

 Mathematics  Number Theory;
 11M26;
 33C45
 EPrint:
 Added a discussion of criteria for deciding whether an equivalence to the Riemann Hypothesis is likely to be useful. Slightly toned down the title