Jensen-Pólya Criterion for the Riemann Hypothesis and Related Problems

Abstract: In this talk, I will summarize forthcoming work with Griffin, Ono, and Zagier. In 1927 Pólya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann’s Xi-function. This hyperbolicity has been proved for degrees $d\leq3$. We obtain an arbitrary precision asymptotic formula for the derivatives $\Xi^{(2n)}(0)$, which allows us to prove thehyperbolicity of $100\%$ of the Jensen polynomials of each degree. We obtain a general theorem which models such polynomials by Hermite polynomials. In the case of Riemann’s Xi-function, this proves the GUE random matrix model prediction for the distribution of zeros in derivative aspect. This general condition also confirms a conjecture of Chen, Jia, and Wang on the partition function.