Let $A$ be a discrete valuation ring of characteristic zero. Let $v$ be the valuation on $A$.
Let $I$ be a finite index set and $d_i$ a positive integer for all $i$ in $I$ and define $$ d:= \sum_{i \in I} d_i.$$
Can we give a nice expression for $$\sum_{i\in I} v(d_i) d_i = \sum_{i\in I} v(d_i^{d_i})$$ in terms of $d$?
If not, can we give a non-trivial bound on the above expression? (Something significantly better than $d^2$.)