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?



Derivation of certain sums "the hard way"
Ramanujan's original paper [*On certain arithmetical functions*](http://ramanujan.sirinudi.org/Volumes/published/ram18.pdf) gives a direct proof. The ideas behind this proof are closely related to the usual modular forms proof, but the words Eisenstein, vector space, modular forms are not mentioned. Indeed Ramanujan says explicitly that his identities "are of course really results in the theory of elliptic functions" and that "the elementary proof of these formulae given in the preceding sections seems to be of interest in itself." 


Briefly, Ramanujan writes down the power series for $P$ (which is $E\_2$), $Q$ (which is $E\_4$) and $R$ (which is $E\_6$), and shows that 
$$ 
\Phi\_{r,s}(x) = \sum\_{n=1}^{\infty} x^n \Big(\sum\_{ab=n} a^r b^s\Big) 
$$ 
can be expressed by polynomials in $Q$ and $R$. Then he multiplies $\Phi\_{0,r}$ and $\Phi\_{0,s}$ and compares the answer with $\Phi\_{1,r+s}$ and $\Phi\_{0,r+s+1}$.


