  We use multiple zeta functions to prove, under suitable assumptions, precise
asymptotic formulas for the averages of multivariable multiplicative functions.
As applications, we prove some conjectures on the average number of cyclic
subgroups of the group ${\mathbb Z}_{m_1}\times\dots\times {\mathbb Z}_{m_n}$
and multivariable averages associated with the LCM function.
