Probability that the k-gcd of products of positive integers is B-friable

In 1849, Dirichlet [5] proved that the probability that two positive integers are relatively prime is 1/ζ(2)1/ζ(2). Later, it was generalized into the case that positive integers have no nontrivial kth power common divisor. In this paper, we further generalize this result: the probability that the gcd of m products of n positive integers is B  -friable is ∏p>B[1−{1−(1−1p)n}m] for m≥2m≥2. We show that it is lower bounded by 1ζ(s) for some s>1s>1 if B>nmm−1, which completes the heuristic proof in the cryptanalysis of cryptographic multilinear maps by Cheon et al. [2]. We extend this result to the case of k  -gcd: the probability is ∏p>B[1−{1−(1−1p)n(1+H1np+⋅⋅⋅+Hk−1npk−1)}m], where Hin=(n+i−1i).