Jensen-Pólya Criterion for the Riemann Hypothesis and Related Problems

When/Where:

October 24, 2017, 3:00 — 3:50 pm at LIT 368.

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 the hyperbolicity of \(100\%\) of the Jensen polynomials of each degree. We obtain a general theorem which models such polynomials by Hermite polynomials. This general condition also confirms a conjecture of Chen, Jia, and Wang.