The University of Arizona

Jensen-Pólya Criterion for the Riemann Hypothesis and Related Problems

Jensen-Pólya Criterion for the Riemann Hypothesis and Related Problems

Series: Algebra and Number Theory Seminar
Location: ENR2 S395
Presenter: Larry Rolen, Georgia Tech

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.

 

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