Mathematics Department Colloquium: Orthogonal polynomial expansions for the Riemann xi function

Colloquium | September 19 | 4:10-5 p.m. | 60 Evans Hall

 Dan Romik, University of California-Davis

 Department of Mathematics

The Riemann xi function is a symmetrized and entire-ized version of the Riemann zeta function, with the property that the Riemann hypothesis is true if and only if all the zeros of the Riemann xi function are real. There is an extensive history of attempts, starting in the early twentieth century (with work by Polya, Turan, De Bruijn and others), to gain insight into the Riemann hypothesis by studying complex-analytic properties of the Riemann xi function, particularly its various representations as a Fourier transform, as a Taylor series, or other "natural" representations. While so far unsuccessful in resolving the main question of interest, these attempts are still quite appealing and continue to attract mainstream attention and lead to new insights, as for example with the recent (2018) proof by Tao and Rodgers of a well-known 1976 conjecture of Newman.

In this talk I will survey some of this history and then describe new results which, following up on a line of investigation started by Turan in the 1950s, support the view that the Riemann xi function may perhaps be best approached by looking at its expansions in several specific families of orthogonal polynomials: the Hermite polynomials first suggested by Turan, and two other less well-known families, the Meixner-Pollaczek polynomials and the continuous Hahn polynomials. I will explain why Turan thought the Hermite expansion was the right tool for the job, and why the new expansions seem in some ways even better.