Riemann $\Xi$-function · Question 1

Workshop introduction

A list of the problems from the workshop on stability, hyperbolicity, and zero localization of functions, held December 5-9, 2011, is presented below. The problems have been loosely categorized for convenience, although many of the problems could be listed under several different headings.

Section context

Let

where

Question context

Let $p_k(z)$ be the polynomials orthogonal with respect to $\Phi$ on $(-\infty,\infty)$. Define

Historical workshop record. Current open/solved status has not been independently reassessed.

Status note in the source

The motivation for this problem is the following. If for all $n\in\mathbb{N}$, $P_n(z,z,\ldots,z)$ vanishes only when $z=i\alpha$, with $\alpha\in\mathbb{R}$, then the Riemann hypothesis is true. Alternatively, if for each $n\in\mathbb{N}$, $Q_n(z,z,\ldots,z)$ is non-zero on the imaginary axis, then the Riemann hypothesis is true.