Improved Averaged Distribution of $d_3(n)$ in Prime Arithmetic Progressions

We say that $d_3(n)$ has exponent of distribution $\theta$ if, for every $\varepsilon>0$, the expected asymptotic holds uniformly for all moduli $q \le x^{\theta-\varepsilon}$. Nguyen proved, following earlier work of Banks, Heath-Brown, and Shparlinski, that after averaging over reduced residue classes $a \bmod q$, the function $d_3(n)$ has exponent of distribution $2/3$. Using the Petrow--Young subconvexity bound for Dirichlet $L$-functions, we improve this to $8/11$ when averaging over residue classes modulo a prime $q$.