Text about connections between complex analysis and partition theory?

I hope this is enough about maths to ask here. As part of my degree I need to do a project consisting of a 7,000-word report on some area of maths (quite a general guideline). I've noticed in studying complex analysis that quite frequently, studying a function's residues winds up giving you an expression for an infinite sum or a generating function for some partition. For instance, there's a well-documented example of using elliptic functions to prove Jacobi's Triple Product.

I also know a little bit about partitions, having read the book Integer Partitions by Andrews and Eriksson and done a previous report summarising a large part of it. So I'd like this project to combine the two areas, and I know -- mostly from casual references -- that it's generally agreed that complex analysis and additive number theory are 'closely related'. The two subjects also seem well-adapted for combining because doing the working out in complex analysis is elegant, but the final answer might not be very exciting, while it's the other way round in number theory, where the working out can be very painful but the final answer pretty neat.

Can you recommend any online source that explores connections between complex analysis and partition theory? If there's an especially good textbook, that would be welcome too. I've noticed Complex Analysis in Number Theory by Anatoly A. Karatsuba, but it doesn't seem like the right fit based on the Amazon description.

EDIT: And if there's a source that links general number theory results with complex analysis, that's cool too. It doesn't have to be partitions. I'm offering a bounty for this now because responses have been extremely scant and it would be really helpful to find a good source.

EDIT 2: I should emphasise that this is a project for my final year of an undergraduate degree. Also, I've found a potentially useful source in the textbook Complex Analysis by Eberhard Freitag and Rolf Busam. Chapter 8 is all about additive number theory. But I'll leave the question open in case there are more sources out there.

• – lhf May 18 '18 at 13:03
• That looks like a good goal to work towards, definitely. – SPS May 18 '18 at 13:40

There's a relatively new body of work authored by M. Merca and myself on so-termed Lambert series factorization theorems which ties the multiplicative functions involved in Lambert series generating functions (an application of analysis is definitely implied here) and the partition function $p(n)$ and restricted partition functions. See the following links for papers (all but one of them is still being reviewed):
Additionally, you can read about our joint work connecting the Euler totient function and the Moebius function with restricted partitions and $p(n)$ (both with Lambert series expansions) here and here, respectively (to be) published this year in the American Mathematical Monthly and the Ramanujan Journal. Additionally, you can check out my recent publication in Acta Arithmetica connecting partitions and general Lambert series expansions here.