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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?

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    $\begingroup$ I upvoted because I had this exact question in mind for a long time, but for some reason I never asked it on MO. $\endgroup$
    – user40023
    Commented Oct 8, 2017 at 15:31
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    $\begingroup$ The analytic meaning is $\mathbb{C}[E_4,E_6]$ contains all the modular forms. What would be the arithmetic meaning of those identities mixing convolution and Dirichlet convolution ? (the simplest example should be $1 \star \chi_4 =\frac{1}{4} \ 1_{\mathbb{Z}^2} \ast 1_{\mathbb{Z}^2}$ which tells about the factorization in $\mathbb{Z}[i]$) $\endgroup$
    – reuns
    Commented Oct 8, 2017 at 19:18

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Ramanujan's original paper On certain arithmetical functions 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 $\Phi_{0,r+s+1}$.

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A really different proof, and the one Zagier refers to, is by Nils Skoruppa: "A quick combinatorial proof of Eisenstein series identities", J. Number Theory 43 (1993), 68--73.

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