On a class of lacunary almost Newman polynomials modulo p and density theorems

TL;DR

This paper studies the reduction modulo p of lacunary polynomials linked to the dynamical zeta function of the beta-shift, exploring their zeroes, factorizations, and connections to modular forms and number theory.

Contribution

It introduces new density theorems and investigates the zeroes and factorizations of these polynomials, connecting them to modular forms and the Langlands program.

Findings

Partial answers to zeroes and factorizations questions using density theorems.

Explicit connections established with modular forms in some cases.

Insights into the distribution of zeroes related to Lehmer's problem.

Abstract

The reduction modulo $p$ of a family of lacunary integer polynomials, associated with the dynamical zeta function $\zeta_{\beta}(z)$ of the $\beta$-shift, for $\beta > 1$ close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in $\mathbb{F}_p$ and their factorizations, via Kronecker's Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These questions arise from the search for the existence of integer polynomials of Mahler measure > 1 less than the smallest Salem number 1.176280. Explicit connection with modular forms (or modular representations) of the numbers of zeroes of these polynomials in $\mathbb{F}_p$ is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.