On the Farey series and outer zeros of partition polynomials (Q1614671)

The Farey series is \[ \mathcal{F}_n = \Big{\{}\frac{h}{k} \in (0,1]: 0 < h \leq k \leq n, \gcd(h,k) = 1\Big{\}}. \] The partition polynomials are given by the formula \[ \wp_n(x) = \sum_{k = 1}^nx^{n(n-k)}\prod_{m=k}^n(1-x^m). \] Quoting the author, ``there is a surprisingly close correspondence between the outer zeros of \(\wp_n(x)\) and the zeros of \(P_n(x) := \prod_{m=1}^n(1-x^m)\), which clearly are the roots of unity \(x = e^{2\pi i h/k}\) with \(h/k \in \mathcal{F}_n\), each occurring with multiplicity \(\lfloor n/k \rfloor\).'' Here the term outer zero refers to zeros outside the unit disk. This close correspondence is illustrated in this paper with one primary result and a number of conjectures backed up by some data and heuristics. We state this main theorem here, recalling that the \textit{partial quotients} of real number \(\theta \in (0,1)\) are the positive integers \(a_i\) in its simple continues fraction representation \[ \theta = \frac{1}{a_1 + \frac{1}{a_2 + \cdots}}. \] Suppose \(\theta \in (0,1)\) is irrational and has bounded partial quotients. If, for all \(n \geq 2\) we have \(\wp_n(x_n) = 0\), where \((x_n)_{n=2}^{\infty}\) is a sequence of complex numbers \(x_n = r_ne^{2 \pi i \theta_n}\) with \(r_n > 1\) and \(\lim_{n \to \infty} \theta_n = \theta\), then \[ r_n < 1 + \frac{1}{n^3} \] for all sufficiently large \(n\).