Curious divisor-like sums Hello everyone
In connection with calculating the Fourier coefficients of some quasi-modular forms which I have been looking at lately, I have come across the following type of sum
$$ S_{a,b}(N) := \sum_{t=1}^{N-1} \ \ \sum_{(n,m) \in I_{N-t,t}} \frac{1}{m^a n^b} $$
where $$ I_{k,l} = \{ \ (m,n) \ \big| \ \ m|k \ , \ n| l \ , \ m>n  \}$$
Note the final condition in $I_{k,l}$ which is of course what prevents this sum from simply being a sum of products of divisor functions. My questions are 
1.)  Can anybody think of a way of calculating this sum rapidly for large $N$. Perhaps by somehow expressing it as a sum of some modified divisor functions or something similar. 
2.) Has series of this type occurred elsewhere in math? 
 A: The sum in question can be rewritten as
$\sum_{m\leq N-1}\sum_{n\leq N-1,m\leq n-1}\frac{1}{m^an^b}\sum_{t\leq N-1,t=0(\bmod m),t=N(\bmod n)}1,$ 
that is
$\sim (N-1)\sum_{m\leq N-1}\sum_{n\leq N-1,m\leq n-1,(m,n)|N}\frac{(m,n)}{m^{a+1}n^{b+1}}.$
Of course, there will come out a so-called "error" term, I haven't check it is really an error.
Sorry, I am not quite familar with the latex on mathoverflow.
A: This function seems related to Percy MacMahon's generalized sum-of-divisors functions. A generating function can be given as follows:
Let $A_k(q) = \sum_n a_{n,k}q^n$ where $a_{n,k}$ is given by the sum $\sum s_1 \cdots s_k$ taken over all ways of writing $k = m_1s_1 + \cdots m_ks_k$ with $0 < m_1 < \cdots < m_k$.
Specifically, $a_{2,k}$ looks similar to what you've written above: You're summing over all ways of partitioning $N$ into two numbers, and summing over the divisors of those numbers... or at least some power of their reciprocals.
I don't see exactly how to write what you've done in terms of these functions (note the condition that you have that $m > n$ is the reverse of the inequality in MacMahon's function i.e. you're summing over the product of the $m_k$ in his function instead of the $s_k$), but they're not too far off.
There is some literature on these functions; there's MacMahon's original paper "Divisors of numbers and their continuations in the theory of partitions", as well as a few papers by George Andrews more recently (one co-authored by me). These generating functions are quasi-modular forms, so it seems to be at least on a similar tack...
