a heuristic, one expects any given prime to have a “probability” $1-O(\exp(-c \log^3 p))$ of having at least one solution, which by the Borel-Cantelli lemma suggests that the Erdős-Straus conjecture is true for all but finitely many $p$. Of course, this is only a heuristic and does not constitute a rigorous argument. (However, one can view the results in [@vaughan], [@els], based on the large sieve, as a rigorous analogue of this type of reasoning.)

From Theorem \[main\] we have the lower bound $\sum_{n {\leqslant}N} f(n) \gg N \log^3 N$. In fact one has the stronger bound $\sum_{n {\leqslant}N} f(n) \gg N \log^6 N$ (Heath-Brown, private communication) using the methods from [@heath]; see Remark \[heath-remark\] for further discussion. Thus, for composite $n$, most solutions are in fact neither of Type I or Type II. It would be of interest to get matching upper bounds for $\sum_{n {\leqslant}N} f(n)$, but this seems to be beyond the scope of our methods. It would of course also be interesting to control higher moments such as $\sum_{p {\leqslant}N} f_{{\operatorname{I}}}(p)^k$ or $\sum_{p {\leqslant}N} f_{{\operatorname{II}}}(p)^k$, but this also seems to unfortunately lie out of reach of our methods, as the level of the relevant divisor sums 