2022/01/10 by Denys Dutykh, Dutykh, Denys, Jean-Louis Verger-Gaugry +1
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2201.03233
openalex publication_date 2022/01/10 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
The reduction modulo p of a family of lacunary integer polynomials,\nassociated with the dynamical zeta function \ζ\β(z) of the\n\β-shift, for \β > 1 close to one, is investigated. We briefly recall\nhow this family is correlated to the problem of Lehmer. A variety of questions\nis raised about their numbers of zeroes in mathbbFp and their\nfactorizations, via Kronecker's Average Value Theorem (viewed as an analog of\nclassical Theorems of Uniform Distribution Theory). These questions are\npartially answered using results of Schinzel, revisited by Sawin, Shusterman\nand Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These\nquestions arise from the search for the existence of integer polynomials of\nMahler measure > 1 less than the smallest Salem number 1.176280. Explicit\nconnection with modular forms (or modular representations) of the numbers of\nzeroes of these polynomials in mathbbFp is obtained in a few cases. In\ngeneral it is expected since it must exist according to the Langlands program.\n