Positive lower density for prime divisors of generic linear recurrences

Olli Järviniemi

paper

Positive lower density for prime divisors of generic linear recurrences

arXiv:2102.04042

Abstract

Let $d \geq 3$ be an integer and let $P \in Z [x]$ 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.

11 pages