Jensen-Pólya Criterion for the Riemann Hypothesis and Related Problems

Series: 
Algebra Seminar
Monday, October 16, 2017 - 15:00
1 hour (actually 50 minutes)
Location: 
Skiles 006
,  
Georgia Tech
,  
Organizer: 
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 $d\leq 3$. 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. This general condition also confirms a conjecture of Chen, Jia, and Wang.