  Recently, Donoso, Le, Moreira and Sun studied the asymptotic behavior of the
averages of completely multiplicative functions over the Gaussian integers.
They derived Wirsing's theorem for Gaussian integers, answered a question of
Frantzikinakis and Host for sum of two squares, and obtained a variant of a
theorem of Bergelson and Richter on ergodic averages along the number of prime
factors of integers. In this paper, we will show the analogue of these results
for co-prime integer pairs. Moreover, building on Frantzikinakis and Host's
results, we obtain some convergences on the multilinear averages of
multiplicative functions over primitive lattice points.
