On the growth and zeros of polynomials attached to arithmetic functions

TL;DR

This paper studies the growth and zero distribution of special polynomials linked to arithmetic functions, extending classical results and applying findings to modular forms and Lie algebra representations.

Contribution

It introduces a new class of polynomials associated with arithmetic functions and extends key classical results to these polynomials, with applications in number theory and representation theory.

Findings

Established bounds on the growth of the polynomials.

Determined the zero distribution of the polynomials.

Applied results to non-vanishing domains of Fourier coefficients.

Abstract

In this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions $g$ and $h$, where $g$ is normalized, of moderate growth, and $0<h(n) \leq h(n+1)$. We put $P_0^{g,h}(x)=1$ and \begin{equation*} P_n^{g,h}(x) := \frac{x}{h(n)} \sum_{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x). \end{equation*} As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind $\eta$-function. Here, $g$ is the sum of divisors and $h$ the identity function. Kostant's result on the representation of simple complex Lie algebras and Han's results on the Nekrasov--Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein's $j$-invariant, and Chebyshev polynomials of the second kind.