1. bookVolumen 17 (2022): Heft 1 (December 2022)
30 Dec 2013
2 Hefte pro Jahr
access type Uneingeschränkter Zugang

On a Class of Lacunary Almost Newman Polynomials Modulo P and Density Theorems

Online veröffentlicht: 31 May 2022
Volumen & Heft: Volumen 17 (2022) - Heft 1 (December 2022)
Seitenbereich: 29 - 54
Eingereicht: 08 Jul 2021
Akzeptiert: 23 Nov 2021
30 Dec 2013
2 Hefte pro Jahr

The reduction modulo p of a family of lacunary integer polynomials, associated with the dynamical zeta function ζβ(z)of the β-shift, for β> 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 𝔽p 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 𝔽p is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.

MSC 2010

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