# 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

## 1 Answer

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.

Between all of these links, you should be able to come up with something plausible for your maths project. I'm also going to be giving a talk at the upcoming George Andrews 80th Birthday conference about these new, more recent connections between additive partition functions and classical multiplicative functions in number theory. If this doesn't do it for you, feel free to email me or post a follow up comment and I will see what else I can come up with in the way of references for your project.

• In particular, you can apply complex contour integrals to the known closed-form Lambert series generating functions for many classical special functions in number theory to obtain the coefficients of these series -- and then compare with our results for these coefficients in the above references. I hope that helps. Sounds like an interesting (Masters?) project. – mds May 24 '18 at 23:17
• I can also put you in touch with a friend who has used the circle method to obtain asymptotic estimates for statistics of special partition functions. – mds May 24 '18 at 23:18
• Thank you for these pointers. At a glance they look a little too advanced for me -- I should have pointed out that this is a fourth-year undergraduate project -- but I'll look at them more carefully in the near future. – SPS May 27 '18 at 11:25
• Have you looked at Hardy and Wright's book? It has many sections on computing asymptotic formulas for average order sums of various arithmetic functions. – mds May 27 '18 at 17:47
• I haven't yet, but I'm interested in proving the asymptotic formula mentioned here: mathoverflow.net/a/43467/532. I'll check out Hardy & Wright, thanks. – SPS May 27 '18 at 18:05