Orthogonal polynomial expansions for the Riemann xi function

Abstract

We study infinite series expansions for the Riemann xi function $\Xi(t)$ in three specific families of orthogonal polynomials: (1) the Hermite polynomials; (2) the symmetric Meixner-Pollaczek polynomials $P_n^{(3/4)}(x;\pi/2)$; and (3) the continuous Hahn polynomials $p_n\left(x; \frac34,\frac34,\frac34,\frac34\right)$. The first expansion was discussed in earlier work by Tur\'an, and the other two expansions are new. For each of the three expansions, we derive formulas for the coefficients, show that they appear with alternating signs, derive formulas for their asymptotic behavior, and derive additional interesting properties and relationships. We also apply some of the same techniques to prove a new asymptotic formula for the Taylor coefficients of the Riemann xi function. Our results continue and expand the program of research initiated in the 1950s by Tur\'an, who proposed using the Hermite expansion of the Riemann xi function as a tool to gain insight into the location of the Riemann zeta zeros. We also uncover a connection between Tur\'an's ideas and the separate program of research involving the so-called De Bruijn-Newman constant. Most significantly, the phenomena associated with the new expansions in the Meixner-Pollaczek and continuous Hahn polynomial families suggest that those expansions may be even more natural tools than the Hermite expansion for approaching the Riemann hypothesis and related questions.Comment: Changes from previous version: typo corrections, added references and other minor improvements to Chapter 4, formattin