  We study upper bounds for the counting function of common zeros of two
meromorphic functions in various contexts. The proofs and results are inspired
by recent work involving greatest common divisors in Diophantine approximation,
to which we introduce additional techniques to take advantage of the stronger
inequalities available in Nevanlinna theory. In particular, we prove a general
version of a conjectural "asymptotic gcd" inequality of Pasten and the second
author, and consider moving targets versions of our results.
