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 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.

中文翻译：

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通用线性递推的素因数的正下密度

令$d \ge 3$为整数，并令$P \in \mathbb{Z}[x]$为d次多项式，其伽罗瓦群为$S_d$。设$(a_n)$是一个非简并线性递归整数序列，其特征多项式为P。我们证明，在广义黎曼假设下，划分序列$(a_n)$的至少一个非零元素的素数集的较低密度为正。