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? 



Curious divisor-like sums
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...


