The sum of the values of the divisor function in arithmetic progressions whose difference is a power of an odd prime

For $D=p^m$, with $p$ a fixed odd prime, $D\leqslant x^{3/8-\varepsilon}$ and $(l,D)=1$, the asymptotic formula
$$ \sum_{\substack{n\leqslant x\\n\equiv l\!\!\!\!\pmod D}}\tau_k(n)=\frac{xQ_k(\log x)}{\varphi(D)}+O\biggl(\frac{x^{1-\varkappa}}{\varphi(D)}\biggr), $$
is proved, where $\tau_k(n)$ is the number of positive integer solutions of $x_1\cdots x_k=n$, $Q_k(z)$ is a polynomial of degree $k-1$ in $z$ with coefficients depending on $k$ and $p$, $\varkappa=\min\{\varepsilon/16,\beta/k^3\}$ with $\beta$ a positive constant depending on $p$, and the constant involved in the order $O$ depends on $k$, $p$ and $\varepsilon$.
The proof relies on an idea of A. A. Karatsuba that permits one to solve this problem by means of a scheme for a ternary additive problem.
Bibliography: 10 titles.