On Additive Divisor Sums and minorants of divisor functions

We establish asymptotic formulae for various correlations involving general divisor functions $d_k(n)$ and partial divisor functions $d_l(n,A)={\sum }_{q\mathrm{\mid }n:q\le n^A}{d}_{l-1}(q)$, 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.