TITLE: A different kind of divisor sums
QUESTION [6 upvotes]: Define, for a number $n$ the function $\mathcal{T}(n)$ which equals the number of different sums $d_1 + d_2,$ where $d_{1, 2}$ are divisors of $n.$ Now, one assumes that $\mathcal{T}(n)$ is approximately $\tau^2(n)/2$ for most $n$ but is there any analytic machinery to analyse the average/maximal behavior of something like this?

REPLY [3 votes]: One thing you can try to use to get estimates for $\mathcal T$ is bounds for the number of divisors $2^k\leq d\leq 2^{k+1}$. Since pairs of divisors from different dyadic ranges have distinct sums, you can get a rough idea for a lower bound of $\mathcal T$. 
The maximum $$\Delta(n)=\max_{k}|\{d|n:2^k\leq d\leq 2^{k+1}\}|$$ has been estimated by Hooley [On a new technique and its application to the theory of numbers, 1977] - something like $$\sum_{n\leq x}\Delta(n)\ll x(\log x)^{4/\pi-1}.$$ Thus, the typical number $n$ has $\Delta(n)=O((\log x)^{4/\pi-1})$ and so has about $(\log n)^{2/3}$ divisors with each from distinct dyadic ranges. Any pair of such divisors has a distinct sum so you get $\mathcal T(n)\gg (\log n)^{4/3}$. This is some way away from your conjecture however...