Positive lower density for prime divisors of generic linear recurrences

TL;DR

This paper proves, assuming the generalized Riemann hypothesis, that the set of primes dividing at least one term of a certain linear recurrence with a polynomial of degree at least 3 and Galois group S_d has positive lower density.

Contribution

It establishes a positive lower density result for primes dividing terms of linear recurrences with specific polynomial characteristics under GRH.

Findings

Primes dividing the sequence have positive lower density.

The result applies to sequences with characteristic polynomials of degree ≥ 3 and Galois group S_d.

Assumes the generalized Riemann hypothesis for the proof.

Abstract

Let $d \ge 3$ be an integer and let $P \in \mathbb{Z}[x]$ 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.