Positive lower density for prime divisors of generic linear recurrences

2021/02/08 by Olli Järviniemi, Järviniemi, Olli
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2102.04042

openalex publication_date 2021/02/08 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

Let d ≥ 3 be an integer and let P ∈ ℤ[x] be a polynomial of degree d whose Galois group is S_d. Let (a_n) be a linearly recuresive sequence of integers which has P as its characteristic polynomial. We prove, under the generalized Riemann hypothesis, that the lower density of the set of primes which divide at least one element of the sequence (a_n) is positive.

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