It is a well-known fact, that one can derive some spectacular identities, e. g.

$\sum^{n-1}_{m=1}\sigma_3(m)\sigma_3(n-m)=\frac {\sigma_7(n)-\sigma_3(n)}{120}$

$\sum^{n-1}_{m=1}\sigma_3(m)\sigma_9(n-m)=\frac {\sigma_{13}(n)-11\sigma_9(n)+10\sigma_3(n)}{2640} $

just by equating several modular forms together. In the book *The 1-2-3 of Modular Forms* Don Zagier writes: "It is not easy to obtain any of these identities by direct number-theroretical reasoning (although in fact **it can be done**)"

Does anybody know how to derive these identities "the hard way" or at least point me to some discussion?