Divisor problem in arithmetic progressions modulo a prime power

We obtain an asymptotic formula for the average value of the divisor function over the integers n≤x in an arithmetic progression n≡amodq, where q=pk for a prime p≥3 and a sufficiently large integer k. In particular, we break the classical barrier q≤x2/3−ε (with an arbitrary ε>0) for such formulas, and, using some new arguments, generalise and strengthen a recent result of R. Khan (2015), making it uniform in k.