Does the Divisor Function $\sigma(n)$ have analogues for other Fuchsian groups? 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.
 A: Looks similar to the elementary formula $(\sum_{k=1}^n k)^2=\sum_{k=1}^n k^3$. Here's why. The function $\sigma_0$ is multiplicative, so 
$$
  \begin{aligned} 
  \sum_{d\mid n} \sigma_0(d) 
  &= \prod_{p^{e_p}\|n} (1+\sigma_0(p)+\sigma_0(p^2)+\cdots+\sigma_0(p^{e_p}))\\
   &= \prod_{p^{e_p}\|n} (1+2+\cdots+(e_p+1))\\
   &= \prod_{p^{e_p}\|n} \frac{(e_p+1)(e_p+2)}{2}.\\
  \end{aligned}
$$
Similarly, since $\sigma_0^3$ is multiplicative,
$$
  \begin{aligned} 
  \sum_{d\mid n} \sigma_0(d)^3 
  &= \prod_{p^{e_p}\|n} (1+\sigma_0(p)^3+\sigma_0(p^2)^3+\cdots+\sigma_0(p^{e_p})^3)\\
   &= \prod_{p^{e_p}\|n} (1^3+2^3+\cdots+(e_p+1)^3)\\
   &= \prod_{p^{e_p}\|n} \left(\frac{(e_p+1)(e_p+2)}{2}\right)^2.\\
  \end{aligned}
$$
Voila, no need for Eisenstein series! And similar results are easily proven.
A: If you are looking for identities involving $\sigma_k$, you could try looking at Duke's article "When is the product of two Hecke eigenforms an eigenform?". For instance, the fact that $E_8=E_4^2$ immediately implies that $$\sigma_7(n)=\sigma_3(n)+120\sum_{m=1}^{n-1}\sigma_3(m)\sigma_3(n-m)$$ by simply computing Fourier coefficients. This work has been extended by a number of authors. (Duke only considered eigenforms for $\mathrm{SL}_2(\mathbb Z)$.) For instance, Emmons and Lanphier (in "Products of an arbitrary number of Hecke eigenforms") considered products of an arbitrary number of eigenforms and Johnson (in "Hecke eigenforms as products of eigenforms") considered eigenforms of higher level (e.g. eigenforms for the Fuchsian group $\Gamma_1(N)$).
In essence, the idea is that whenever you have a relation involving Hecke eigenforms, you get an elementary number theoretic identity by considering the Fourier coefficients.
