<abstract xmlns="http://www.w3.org/1999/xhtml">

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

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.

On a Class of Lacunary Almost Newman Polynomials Modulo <italic>P</italic> and Density Theorems

Univ. Grenoble Alpes, Univ. Savoie Mont Blanc

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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

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...

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

Published Online: May 31, 2022

Page range: 29 - 54

Received: Jul 08, 2021

Accepted: Nov 23, 2021

DOI: https://doi.org/10.2478/udt-2022-0007

Keywords
lacunary integer polynomial, zeroes, factorization, Lehmer’s problem, Chebotarev density theorem, Frobenius density theorem, number of zeroes modulo

© 2022 Denys Dutykh et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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

Published Online: May 31, 2022

Page range: 29 - 54

Received: Jul 08, 2021

Accepted: Nov 23, 2021

DOI: https://doi.org/10.2478/udt-2022-0007

Keywordslacunary integer polynomial, zeroes, factorization, Lehmer’s problem, Chebotarev density theorem, Frobenius density theorem, number of zeroes modulo

© 2022 Denys Dutykh et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
lacunary integer polynomial, zeroes, factorization, Lehmer’s problem, Chebotarev density theorem, Frobenius density theorem, number of zeroes modulo