Orthogonal polynomial expansions for the Riemann xi function in the Hermite, Meixner--Pollaczek, and continuous Hahn bases
Volume 200 / 2021
Abstract
Pál Turán in the 1950s proposed to use the expansion of the Riemann xi function in the Hermite polynomials as a tool to gain insight into the location of the zeros of the Riemann zeta function. In this paper we follow up and expand on Turán’s ideas in several ways by considering 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 $p_n\bigl (x; \frac 34,\frac 34,\frac 34,\frac 34\bigr )$. For each of the three expansions we derive asymptotic formulas for the coefficients and prove additional results. We also apply some of the same techniques to prove a new asymptotic formula for the Taylor coefficients of the Riemann xi function, and uncover a previously unnoticed connection between the Hermite expansion of $\Xi (t)$ and the separate program of research involving the de Bruijn–Newman constant.
