Asymptotic analysis on the Riemann xi function and its orthogonal polynomial expansions

Dan Romik
University of California, Davis (UC Davis)

The methods in the toolkit of asymptotic algebraic combinatorics draw on, and are applicable to, many different branches of mathematics. In this talk I will show how some of those same methods can be applied to study one of the most classical objects in mathematics, the Riemann zeta function, and its close cousin the Riemann xi function. One motivation is a classical attack on the Riemann hypothesis originated by Turan, who in the 1950s argued that the question of whether the Riemann zeta zeros lie on the critical line is best addressed by expanding the Riemann xi function in Hermite polynomials. I will explain why the Hermite polynomials are not the only interesting family of orthogonal polynomials in which it is worth expanding the Riemann xi function, and that in particular there are two other families for which the expansion leads to good behavior and can be analyzed to a good level of accuracy.

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