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 $k\ge 7$.
Categories: math.NT