For the Riemann xi-function, the Jensen polynomials J^{d,n} are hyperbolic whenever n^3 log^2(n+2) ≥ K d^5, and their scaled zeros converge to Wigner's semicircle law in this joint limit.
Finite free convolutions of polynomials
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abstract
We study three convolutions of polynomials in the context of free probability theory. We prove that these convolutions can be written as the expected characteristic polynomials of sums and products of unitarily invariant random matrices. The symmetric additive and multiplicative convolutions were introduced by Walsh and Szeg\"o in different contexts, and have been studied for a century. The asymmetric additive convolution, and the connection of all of them with random matrices, is new. By developing the analogy with free probability, we prove that these convolutions produce real rooted polynomials and provide strong bounds on the locations of the roots of these polynomials.
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A new hyperbolicity wedge and a joint semicircle limit for Jensen polynomials of Riemann's $\xi$-function
For the Riemann xi-function, the Jensen polynomials J^{d,n} are hyperbolic whenever n^3 log^2(n+2) ≥ K d^5, and their scaled zeros converge to Wigner's semicircle law in this joint limit.