  We study the average value of the divisor function $\tau(n)$ for $n\le x$
with $n \equiv a \bmod q$. The divisor function is known to be evenly
distributed over arithmetic progressions for all $q$ that are a little smaller
than $x^{2/3}$. We show how to go past this barrier when $q=p^k$ for odd primes
$p$ and any fixed integer $k\ge 7$.
