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Zeros of Jensen polynomials and asymptotics for the Riemann xi function

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arxiv 2007.13582 v2 pith:F3KS6FMF submitted 2020-07-27 math.NT

Cormac O'Sullivan This is my paper · ORCID
classification math.NT
keywords jensenpolynomialscriterionfunctionriemannzerosassociatedasymptotic
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abstract

The classical criterion of Jensen for the Riemann hypothesis is that all of the associated Jensen polynomials have only real zeros. We find a new version of this criterion, using linear combinations of Hermite polynomials, and show that this condition holds in many cases. Detailed asymptotic expansions are given for the required Taylor coefficients of the xi function at $1/2$ as well as related quantities. These results build on those in the recent paper of Griffin, Ono, Rolen and Zagier.

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  1. A new hyperbolicity wedge and a joint semicircle limit for Jensen polynomials of Riemann's $\xi$-function

    math.NT 2026-08 accept novelty 8.0 of 10

    For the Riemann xi-function, the Jensen polynomials J^{d,n} are hyperbolic whenever n^3 log^2(n+2) ≥ K d^5, and their scaled zeros converge to Wigner's semicircle law in this joint limit.

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