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Number Theory Seminar: Jensen-Pólya Criterion for the Riemann Hypothesis and Related Problems

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Speaker: Larry Rolen, Vanderbilt University
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Location: 3106 MSB
Start time: Mon, May 13 2019, 2:10PM

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.