Seminar

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I will discuss an approach to studying the Riemann zeta function and its close cousin, the Riemann xi function, using orthogonal polynomials. The idea originates with Turan, who in the 1950s suggested expanding the Riemann xi function in the basis of Hermite polynomials as a way of gaining insight into the location of the nontrivial zeros of the Riemann zeta function. In recent work I considered the same Hermite expansion and two additional expansions in more exotic families of orthogonal polynomials, the Meixner-Pollaczek polynomials and the continuous Hahn polynomials. I will explain what makes these particular expansions interesting to study, and how expansions of this type relate to other well-studied questions related to the Riemann zeta function, such as Newman's conjecture about the nonnegativity of the De Bruijn-Newman constant, recently proved by Rodgers and Tao.

Orthogonal Polynomial Expansions For The Riemann Xi Function