Let \(d_k(n)\) denote the k-fold divisor function. For a wide range of large q the expected bound \(\begin{aligned} \sum _{\begin{array}{c} n\le x\\ {n\equiv a(q)} \end{array}}d_k(n)-\text { main term }\approx \sqrt{\frac{x}{q}} \end{aligned}\) is shown to be true in an average sense - for all k. This generalises the work of Pongsriiam and Vaughan [18] who studied \(k=2\) , and answers the work of Rodgers and Soundararajan [19], 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 [8] to study the unsmoothed problem.
