Positive lower density for prime divisors of generic linear recurrences

Abstract

Let d = 3 be an integer and let P ? Z[x] be a polynomial of degree d whose Galois group is Sd. Let (a(n)) be a non-degenerate linearly recursive sequence of integers which has P as its characteristic polynomial. We prove, under the generalised Riemann hypothesis, that the lower density of the set of primes which divide at least one non-zero element of the sequence (a(n)) is positive.