It can be shown, by using the fact that $E_4(z)^2 = E_8(z)$ (where the $E_k$ are Eisenstein series), that $$\sigma_7(n) = \sigma_3(n) + 120\sum_{0 < m < n}\sigma_3(m -n)\sigma_3(n)$$ where $\sigma_k(n) := \sum_{d|n} d^k$. Is there a more elementary proof of this, and is there a simple reason one should expect such identities to hold?
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2 Answers
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I have asked a question about a different proof of such identities here: Derivation of certain sums "the hard way", but I would say that the non-modular proofs are much less elementary than the modular ones.
To answer your second question, we should expect these identities to hold, because Eisenstein series have the coefficients of the form $\sigma_k(n)$. If you combine this with some properties of modular forms (multiplying a form of weight $k$ and $l$ gives you a form of weight $k+l$ and that the space of mod. forms of given weight has a given dimension), you can see that some identities must arise between them. This is not only the case with the $\sigma_k$, other arithmetic functions also have modular identities.
The identity you gave above is a perfect example. You know, that both $E_4^2$ and $E_8$ are forms of weight 8, but the dimension of $M_8(\Gamma)$ is 1. Thus, one is a multiple of the other (actually, $E_4^2=E_8$) and you have your identity.
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There is paper that gives a quite complete account of elementary methods for evaluating many kind of sum-of-divisors convolution formulae, including the formula you mention: Huard J.G., Ou Z.M., Spearman B.K. and Williams K.S., ‘Elementary evaluation of certain convolution sums involving divisor functions’, in Number Theory for the Millenium II (A.K. Peters, Natick, MA, 2002), pp. 229–274.
