Define $\tau_k = 1^{*k}$ where $*$ is the Dirichlet convolution. Then it's known that $$\sum_{n \le x} \tau_k(n) = x P_k(\log x) + O(x^{1-1/k} \log^{k-2}(x))$$ where $P_k$ has leading coefficient $\frac{1}{(k-1)!} x^{k-1}$.
Apparently this can be proven through applying the Dirichlet Hyperbola Method with a threshold of $\sqrt[n]{x}$. Does anyone have a write up of this proof, or a link to such a write up?