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# Larry Rolen – Vanderbilt University

## October 29, 2018 @ 4:30 pm - 5:30 pm

**Jensen-Pólya Criterion for the Riemann Hypothesis and Related Problems**

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 less than or equal to 3. We obtain an arbitrary precision asymptotic formula for the derivatives of Xi, 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.