  We prove some distribution results for the $k$-fold divisor function in
arithmetic progressions to moduli that exceed the square-root of length $X$ of
the sum, with appropriate constrains and averaging on the moduli, saving a
power of $X$ from the trivial bound. On assuming the Generalized Riemann
Hypothesis, we obtain uniform power saving error terms that are independent of
$k$.
  We follow and specialize Y.T. Zhang's method on bounded gaps between primes
to our setting. Our arguments are essentially self-contained, with the
exception on the use of Deligne's work on the Riemann Hypothesis for varieties
over finite fields. In particular, we avoid the reliance on Siegel's theorem,
leading to some effective estimates.
