In the 1960s, Hasse studied the frequency, i.e., the Dirichlet density, of prime divisors of sequences V = (a n + 1)n≥0 with a ∈ Z. Hasse’s study involved the multiplicative order of a modulo primes p. Indeed, p divides some term Vn if and only if ordp(a) is even. With a new approach, Pieter Moree gave a more general description of this problem in a 2005 paper.
Let m, d ∈ N and g = (g1, . . . , gm) ∈ Qm. Using Chebotarev’s density theorem, we study the asymptotic behavior of the counting function for primes p ≤ x, with x > 1, such that d divides lcm(ordp(gi) : 1 ≤ i ≤ m). Although a formula for the Dirichlet density is obtained, it is not easily computable. Following Moree’s method, who completed the case m = 1, we find a closed-form of the density for any m ≥ 1 and with some assumptions on g and d.
We apply these theoretical calculations to find the Dirichlet density of prime divisors of sequences which are generalizations of Hasse’s sequences.
