  We establish asymptotic formulae for various correlations involving general
divisor functions $d_k(n)$ and partial divisor functions
$d_l(n,A)=\sum_{q|n:q\leq n^A}d_{l-1}(q)$, where $A\in[0,1]$ is a parameter and
$k,l\in\mathbb{N}$ are fixed. Our results relate the parameter $A$ to the
lengths of arithmetic progressions in which $d_k(n)$ is uniformly distributed.
As applications to additive divisor sums, we establish new lower bounds and a
new equivalent condition for the conjectured asymptotic. We also prove a
Tauberian theorem for general additive divisor sums.
