On multivariable averages of divisor functions

We deduce asymptotic formulas for the sums∑n1,…,nr≤xf(n1⋯nr)and∑n1,…,nr≤xf([n1,…,nr]), where r≥2 is a fixed integer, [n1,…,nr] stands for the least common multiple of the integers n1,…,nr and f is one of the divisor functions τ1,k(n) (k≥1), τ(e)(n) and τ⁎(n). Our formulas refine and generalize a result of Lelechenko (2014). A new generalization of the Busche-Ramanujan identity is also pointed out.