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.

中文翻译：

---

黎曼 xi 函数的 Jensen 多项式的零点和渐近线

黎曼假设的 Jensen 的经典标准是所有相关的 Jensen 多项式都只有实数零点。我们找到了该标准的新版本，使用 Hermite 多项式的线性组合，并表明该条件在许多情况下都成立。给出了 xi 函数在 1/2 处所需的泰勒系数以及相关量的详细渐近展开式。这些结果建立在 Griffin、Ono、Rolen 和 Zagier 最近的论文中。