I have been reading about the divisor function $\sigma = 1 \ast 1$ and proved an elementary identity:

$$ \Big[\sum_{d|n} \sigma_0(d)\Big]^2 = \sum_{d|n} \sigma_0(d)^3$$

Here $\sigma_0 = \sum_{d|n} 1$ counts the divisors of $n$. Could be the first of many such kind of identity? I wouldn't even be sure where to look for identities for $\sigma_k$.

Perhaps you could prove such a result with Eisenstein series:

$$ G_k(z) = - \frac{B_k}{k} + \sum_{n=1}^\infty \sigma_{k-1}(n)q^n = \frac{(2\pi i)^k}{(k-1)!} \times \frac{1}{2}\sum \frac{1}{(m \tau + n)^k}$$

However $G_0$ is not a meaningful Eisenstein series.

Could the divisor function have analogues for other Fuchsian groups? The Ramanujan sum

$$ \sum_{(a,q)=1} e^{2\pi i \frac{a}{q}} = \mu(q)$$

related to the Mobius function. This certainly has analogues for other Fuchsian groups.