A Montgomery-Hooley theorem for the k-fold divisor function

Let $d_k(n)$ denote the $k$-fold divisor function. For a wide range of large $q$ the expected bound $$\sum _{\genfrac{}{}{0px}{}{n\le x}{n\equiv a(q)}}{d}_k(n)-\text{ main term }\approx \sqrt{x\mathrm{/}q}$$ is shown to be true in an average sense – for all $k$. This generalises the work of Pongsriiam and Vaughan [15] who studied $k=2$, and answers the work of Rodgers and Soundararajan [17], who used the asymptotic large sieve to study a smoothed version of the problem. We use a circle method approach as developed by Goldston and Vaughan [7] to study the unsmoothed problem.