What can you say about a thirty-two-year-old mathematician who died? That he loved numbers and equations. That he had a mysteriously intimate understanding of infinite numerical processes (infinite sums, infinite products, infinite continued fractions, and the like). That to the mathematicians of England, his ideas seemed to spring from nowhere — while he himself said that his ideas came from a goddess.
The collaboration that took place in the years 1914–1919 between the Indian mathematician Srinivasa Ramanujan and the English mathematician G. H. Hardy — perhaps the most famous collaboration in the history of mathematics — is the subject of the 1991 book The Man Who Knew Infinity by Robert Kanigel. It’s also the subject of a new film by the same name, which I’ll say more about next month. Today I want to show you three “flowers from Ramanujan’s garden” (to steal a metaphor from Freeman Dyson), with some puzzles strewn along the way, and to explain why Kanigel’s choice of title makes sense. I’ll also speculate a bit about where mathematical ideas come from, and here the case of Ramanujan can be instructive, not because he was typical but because he was such an outlier; it’s hard to think of a parallel example of someone who came up with so many beautiful ideas but had so much trouble leading others to the wellsprings of his inspiration. (His early death was certainly the greatest and most final source of this trouble, but it wasn’t the only one.)
FROM INDIA TO ENGLAND
First, let’s settle the question of how you should say his name. If English is your first language, you’re probably wondering whether to pronounce it as Ra-MA-nu-jan or as Ra-ma-NU-jan. The answer? Neither! You should give all four syllables equal stress. Or at least, that’s a first step toward saying the name correctly. The Wikipedia article on Ramanujan has a sound-clip of a native Tamil speaker pronouncing the name slowly and pedantically; I don’t know what a fluid Tamil utterance of the name would sound like.
Ramanujan was born in the the state of Tamil Nadu in southern India to a musical family. The Carnatic music he heard as a child places a lot of emphasis on rhythm, and some have speculated that this fed Ramanujan’s love of arithmetical pattern. He did not have a mentor, but he did have access to books, and a book on mathematics by G. S. Carr seems to have influenced him deeply, in ways both good and bad. According to Hardy, Carr’s book was not much more than a collection of formulas, and this became Ramanujan’s image of what math should be. (For a more sympathetic account of Carr’s book, see Amithab Sen’s essay.) In Ramanujan’s notebooks, the equals signs featured so prominently in Carr’s book would become amazing bridges between the outcomes of entirely different (often infinite) arithmetic processes — but, as in Carr’s book, the trusses and support-beams of reasoning that held these bridges up would be hidden from view.
Another reason for Ramanujan’s gnomic style is that paper was scarce in India; he worked by writing on slate or drawing in sand, and committed to his notebooks only his final conclusions, not the processes that led him to them.
The young Ramanujan had philosophical interests as well as mathematical ones, and the concept of the infinite gripped him. One of his earliest explorations was the study of infinite nested radicals like . This particular sum happens to equal 3, but in many of Ramanujan’s equations, both the left and right hand side are infinite expressions, and the most intriguing ones are the equations in which the two sides have very different character — one being an infinite sum and the other being an infinite product, say. Talk about comparing apples and oranges!
Ramanujan came to Hardy’s attention when the younger mathematician, then a college dropout working as a clerk, wrote his now-famous letter of introduction. It began “I beg to introduce myself to you as a clerk in the Accounts Department of the. Port Trust Office at Madras” and went on to present a number of claims, including one that would have been easy to dismiss as outright crackpottery: the formula 1+2+3+… = −1/12. This formula has recently been the subject of some irresponsible boggle-mongering on the Web. If you’re encountering this formula for the first time, some skepticism or even outright disbelief is in order. How can a sum of positive numbers be negative? And how can a sum of whole numbers be a fraction? Taken literally, the equation is nonsense. I’ll say that last bit louder: If you interpret “+…” and “=” in the usual, way, the equation is nonsense!
But there’s total nonsense and then there’s the kind of nonsense that hints at new forms of sense, and fortunately for Ramanujan, Hardy knew enough advanced mathematics to recognize the equation as an instance of the latter. In a future blog, I may talk about this equation in more detail, and the hidden role played by the Riemann zeta function, and about how there are different kinds of nonsense in math, some more useful than others. In the meantime, for a sensible explanation of what the equation “1+2+3+4+…=−1/12” is and isn’t, see Evelyn Lamb’s essay “Does 1+2+3… really equal −1/12?”
Hardy recognized that Ramanujan, proceeding in his own self-taught way and expressing himself loosely, was arriving at some of the same conclusions that researchers in the European mainstream had reached. More importantly, Ramanujan was arriving at other conclusions that were entirely novel; it was clear that the European mainstream had as much to learn from him as he had to learn from it. Hardy invited Ramanujan to work with him at Cambridge University, and thus began one of the great mathematical partnerships in history. As Hardy wrote in his famous book “A Mathematician’s Apology”, his collaboration with Ramanujan was “the one romantic incident in my life”.
Ramanujan promised that he would reveal his methods to Hardy, but it quickly became clear to the older mathematician that Ramanujan’s quicksilver way of thinking was hard to divert into channels of rigorous reasoning. He was, in many ways, a throwback to the 18th century, even as his work presaged mathematical advances of the 20th and 21st centuries. He was a kindred spirit to his great predecessor the mathematician Leonhard Euler, whose motto was “My pen is a better mathematician than I am” and who often pushed mathematical notation into places it was never intended to go. Ramanujan worked in the same style as Euler, but mainstream mathematicians had adopted a more rigorous approach to infinite series, and in this new framework, a formula like 1+2+3+4+… = −1/12 was literally false; whatever its poetic truth might be, you couldn’t use it as a springboard for further conclusions. This tension between Ramanujan’s way of thinking and the “modern” approach to the infinite (to which Hardy ardently subscribed) was a major frustration for both collaborators. Still, they managed to bridge this divide, producing in a few short years some pioneering mathematics.
PARTITIONS OF NUMBERS
A large part of Ramanujan’s work concerned a branch of mathematics founded by Euler called partitio numerorum, the study of partitions of numbers. This subfield of number theory is about the different ways in which whole numbers can be written as sums of 1’s, 2’s, 3’s, etc. You can think of this as a problem about the different ways you could pay for a purchase if you had an unlimited supply of 1-cent coins, 2-cent coins, 3-cent coins, etc.
In the number-partition game, we want to write some number n as a sum of one or more positive integers, where two sums are to be counted as the same if they’re just re-arrangements of each other. For instance, 4 can be written as 1+1+1+1, or as 2+1+1 (which counts as the same as 1+2+1 and 1+1+2), or as 3+1 (which counts as the same as 1+3), or as 2+2, or as just 4. (“4” doesn’t actually look like a sum, but we agree to think of it as a sum with only one term, because it turns out that allowing partitions consisting of only one term is more natural in applications such as counting ways to make payment using various denominations of coins; and besides, it leads to prettier results.) So the partitions of 4 are 1+1+1+1, 2+1+1, 3+1, 2+2, and 4. Since there are 5 such sums, we say that the number of partitions of the number 4 is 5. That is, if you’ve got an inexhaustible supply of 1 cent coins, 2 cent coins, 3 cent coins, etc., and you want to pay for something that costs 4 cents, there are 5 combinations of coins you can slap onto the counter to make your purchase. We write the number of partitions of n as p(n); so for instance, we’ve just shown that p(4) = 5.
It’s customary (and sensible) to set p(0) = 1. After all, if you want to pay for something that’s free, there’s exactly one way to pay for your purchase: just hand over 0 coins of each denomination and exit the store smiling.
The partition function p(n) is subtle:
- You can check that p(1) = 1, p(2) = 2, p(3) = 3, and p(4) = 5, which might lead you confidently guess that p(5) will be 8, the next Fibonacci number, but nope: p(5) = 7.
- Okay, but having noticed that p(5) is 7 and p(6) is 11, you might optimistically hope that p(7) (like p(2), p(3), p(4), p(5), and p(6)) will be prime, but nope: p(7) is 15.
- After recovering from that setback, you might forlornly hazard the guess that p(8) should at least be odd (like p(3), p(4), p(5), p(6), and p(7)), but nope: p(8) is 22.
So this sequence 1,2,3,5,7,11,15,22,… doesn’t reveal its secrets right away. In contrast, there’s an extremely simple formula for the number of “ordered partitions” (often called “compositions”) of n, which are like partitions of n except that now order does count (for instance, 1+2 is not the same composition as 2+1); can you guess the formula (by computing values and noticing a pattern), and, having done so, can you explain why the formula works? (See End Note #1.)
My first serious exposure to the sequence 1,2,3,5,7,11,15,22,… came from chapter 19 of Hardy & Wright’s great book “An Introduction to the Theory of Numbers”; I found it very inspiring. You can also learn about the subject from George Andrews’ books or Bruce Berndt’s notes, listed in the References at the end of the essay.
One thing I love about the theory of partitions is that it’s rife with propositions of the form “For all positive integers n, there are exactly as many partitions of n with Property X as there are partitions of n with Property Y.” This wouldn’t be so surprising if X and Y looked similar, but often they look quite different. I also like the fact that propositions in this genre, called “partition identities”, span the gamut of difficulty. Some of them are easy to prove; others are fiendishly difficult. Let me give three examples of this.
Proposition A: For all n, the number of partitions of n containing at most 3 parts equals the number of partitions of n with no parts exceeding 3. (Example: The partitions of 4 containing at most 3 parts are 2+1+1, 3+1, 2+2, and 4; the partitions of 4 with no part exceeding 3 are 1+1+1+1, 2+1+1, 3+1, and 2+2. In both cases, we get four partitions.)
Proposition B: For all n, the number of partitions of n in which no parts are repeated equals the number of partitions of n in which all parts are odd. (Example: The partitions of 4 in which no parts are repeated are 3+1 and 4; the partitions of 4 in which all parts are odd are 1+1+1+1 and 3+1. In both cases, we get two partitions.)
Proposition C: For all n, the number of partitions of n in which no parts are repeated or differ by 1 equals the number of partitions of n in which all parts are adjacent to a multiple of 5, i.e., all parts end with the digit 1, 4, 6, or 9. (Example: The partitions of 4 in which no parts are repeated or differ by 1 are 3+1 and 4; the partitions of 4 in which all parts are adjacent to a multiple of 5 are 1+1+1+1 and 4. In both cases, we get two partitions.)
Propositions A and B were both known to Euler. Proposition A is much the easier of the two, and makes a great problem for a young mathematician. (Hint: try to find a pictorial way to represent partitions. If you get stuck, peek at End Note #2.) Proposition B is harder, but still within reach of a persevering math enthusiast. (Here’s a clue that, considered in the right light, might be helpful: check that the number of partitions of n into unequal powers of 2 equals the number of partitions of n into odd powers of 2. If you get stuck, peek at End Note #3.)
Intimately related to both the infinite sum on the left and the infinite product on the right is the Rogers-Ramanujan continued fraction
this is defined as the limiting value of the sequence , , , …
Although the Rogers-Ramanujan identities were discovered over a century ago, we’re still learning new things about them.
The number 5 turns up in other work of Ramanujan. Take a look at this table of the numbers p(n) with n running from 0 up to 49, arranged in 10 rows of length 5.
I’ve marked the numbers that are divisible by 5 with the color red. You should instantly see that one of the columns in this table is completely red. We don’t know whether Ramanujan drew a table like this, but he did notice that the numbers p(4), p(9), p(14), p(19), p(24), p(29), … are divisible by 5. He also noticed similar patterns when you look for value of p(n) that are multiples of 7: whenever n is “congruent to 5 mod 7” (that is, whenever n leaves a remainder of 5 when divided by 7), p(n) is congruent to 0 mod 7 (that is, p(n) leaves a remainder of 0 when divided by 7, or putting it more plainly, p(n) is divisible by 7). Likewise, when n is congruent to 6 mod 11, p(n) is congruent to 0 mod 11. Somehow these “Ramanujan congruences” (which are peculiar to the numbers 5, 7, and 11) had escaped the attention of Euler, Sylvester, and all the others who had worked on number partitions for more than a century. Keeping in mind that Ramanujan’s mother was a singer in the Carnatic tradition, I can’t help wondering whether early exposure to music in quintuple and septuple meter might have predisposed Ramanujan to notice patterns like this.
Of course, noticing a pattern is one thing; proving it is another; and understanding it in a deep way is yet a third thing. Ramanujan was able to prove the multiple-of-5 and multiple-of-7 facts, and after Ramanujan’s death Hardy was able to prove the multiple-of-11 fact building on Ramanujan’s unpublished papers. But it wasn’t until 2011 that a fully satisfying conceptual explanation of “What’s so special about 5, 7, and 11?” was given.
RAMANUJAN’S PARTITION FORMULA
One of Ramanujan’s greatest contributions to the theory of partitions was a formula for p(n). A useful consequence of Ramanujan’s formula was an asymptotic formula for p(n), independently rediscovered by Uspensky a few years later: When n is large, p(n) is close to P(n), where . When I say that they’re close, I mean that p(n) / P(n) converges to 1 as n goes to infinity; that is, the relative error of the approximation goes to 0 percent. For instance, p(200) = 3,972,999,029,388 while P(200) is approximately 4,100,251,432,188: an error of only about 3 percent.
But Ramanujan didn’t stop there. He found an infinite progression of better and better approximations to p(n) that allow you to compute p(n) exactly, for any given n. He and Hardy showed that when k is about the square root of n, the kth Ramanujan approximation to p(n) differs from the true value of p(n) by less than 1/2, so to determine the exact value of p(n), you can compute the kth approximation and round to the nearest integer. Actually, to say “when k is about the square root of n” overstates the computational difficulty of the method — for instance, when n is 200, taking k=5 will do.
I mention the case n=200 because, as a way of checking that Ramanujan’s formula was valid, Hardy and Ramanujan had Percy MacMahon compute p(200) “the old-fashioned way” (which I’ll discuss in next month’s “movie review”), and then they computed it with their new formula: in both cases, the answer 3,972,999,029,388 was obtained. You can see this dramatic verification in the movie, sort of.
Hardy and Ramanujan’s proof of the formula hinged on the art of understanding how functions “behave near infinity” — a subtle notion, because infinity isn’t a number, and no finite number is anywhere close to being infinite. But when we say that n2 is negligible compared to 2n when n is large, even though both of them increase to infinity as n does, we’re engaged in the kind of thinking Hardy and Ramanujan needed in order to prove their formula. So in that sense, Hardy and Ramanujan were both mathematicians who “knew infinity”: they knew how to work with it. But they knew in different ways. Hardy’s way of knowing infinity was more cautious, more painstaking, and more in line with what other mathematicians of that period were doing. Ramanujan was bolder, with an insouciance that by rights should have gotten him into trouble more often than it did. In many cases Ramanujan arrived at correct conclusions about infinite sums, infinite products, and infinite continued fractions without using correct forms of reasoning; his intuition had somehow become tuned to mathematical reality, so that even when he “went off-road”, his instincts didn’t lead him into swamps of falsehood. He wasn’t always right, but he wasn’t often wrong.
Robert Kanigel tells me that, as he learned about Ramajunan and Hardy, he developed an image “of an explorer heading into the Pyrenees, for example, and needing a guide. He seeks someone who knows the countryside, the terrain, the surprising twists of topography. So it was always ‘infinity’ as a kind of terrain one might know, that Ramanujan ‘knew’, that was the basis for the title.”
WHERE DID RAMANUJAN’S IDEAS COME FROM?
How did Ramanujan achieve his intuitive grasp of infinite sums, products, and continued fractions? To some of the people who asked Ramanujan where his ideas came from, he gave credit to his hometown goddess Namagiri, a local manifestation of Lakshmi, the Hindu goddess of good fortune. If any of you know more about those remarks and the context in which Ramanujan made them, I’d be grateful for details. It’s possible that Ramanujan was being completely sincere in saying that some of his formulas arrived from the goddess in their final form, but it’s also possible that he had a more complex view of his creative processes, in which external inspiration and internal effort both played a role. I’m reminded of a Ph.D. thesis whose introduction thanked “my savior Jesus Christ for inspiration, particularly with regard to Lemma 4.2” or something like that (I can’t dig up a reference for this, but maybe one of you can!).
There’s also a striking quotation from Ramanujan on the Wikipedia page about Namagiri: “While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing.” Trouble is, I’m having difficulty finding a primary source (Wikipedia’s source is a 2011 book about yoga!). One possibility is that the quotation originated in David Leavitt’s “The Indian Clerk” and that someone posted it on the web, not realizing that the book is a novel based on Ramanujan’s life as opposed to a factual biography. But maybe the quote is legitimate. Can any of you readers help solve this mystery?
John Baez points out that if a god’s existence could be proved by miracles, Namagiri would be the best-attested god around, by virtue of the miraculous insights she gave Ramanujan. I’m steeped in the Western agnostic tradition, so of course I’m skeptical about a goddess having contributed to Ramanujan’s oeuvre. But the fantasist in me can’t help hoping it’s true. I mean, Namagiri, apart from being divine and all that, is sort of a local; if Ramanujan’s formulas are a sample of what a farm-league Hindu deity knows, imagine what big-city Lakshmi could teach us!
In this essay I’ve tried to blend engaging story-telling with journalistic accuracy (if not academic punctiliousness), but part of me feels that, in painting a portrait of Ramanujan as a savant, I may have strayed too far from the opinion of one of the people most qualified to weigh in on the question. Hardy wrote: “I have often been asked whether Ramanujan had any special secret; whether his methods differed in kind from those of other mathematicians. I cannot answer these questions with any confidence or conviction; but I do not believe it. My belief is that all mathematicians think, at bottom, in the same kind of way, and that Ramanujan was no exception.” Hardy went on to mention Ramanujan’s unusual skill with calculation and his extraordinary power of memory, but averred that in neither department was he freakish. “It was his insight into algebraic formulae, transformations of infinite series and so forth, that was most amazing. On this side most certainly I have never met his equal, and I can compare him only with Euler or Jacobi.”
EXPERIMENTS WE CAN DO (AND ONE WE CAN’T)
One of the great things about this new century is the promise that, for the first time, researchers can find out what we mathematicians are doing on a neurological level when we “go off” into our alternative reality. Is the process of learning to do math partly a matter of repurposing regions of the brain? If so, which regions? When I was in graduate school, I heard people talk about there being “ear-mathematicians” (algebraists) and “eye-mathematicians” (geometers); this dichotomy suggests that the sensory region of the brain are the ones most likely to play a role in creative mathematics. (Ramanujan himself may have been a “mouth-mathematician”; he once claimed he woke up to find a formula written on his tongue by the goddess!) Personally, I’d broaden “sensory” to “sensorimotor”; when I enter into the voluntary semi-hallucinatory state called “thinking about mathematical objects”, I feel that mathematical objects are things I can actually manipulate and act upon.
Ramanujan’s case may support a different hypothesis: that doing math in a deep way involves the social part of the brain that recognizes individuals. The famous story of Ramanujan and his instant recollection of a pretty property of the number 1729 (which I won’t repeat here, but is worth looking up if it’s unfamiliar to you) led mathematician J. E. Littlewood to speculate that Ramanujan was friends with each of the positive integers. The mathematician John von Neumann once said “In mathematics you don’t understand things; you just get used to them.” That certainly would seem to apply to infinity. But could the same not be said of other humans, who in some sense are ultimately unknowable to us (even when we count them as friends)? And if so, might the same parts of the brain be involved? Here’s an experiment we can’t do, but is intriguing to imagine anyway: If you could put Ramanujan in a functional MRI machine, ask him to think about his friend Hardy, and then ask him to think about the number 1729, would the two brain regions that lit up be suspiciously close together?
I have a personal interest in such questions, since I’ve suffered from a form of face-blindness for most of my life. (In fact, while I was writing this essay at a cafe, the father of one of my children’s friends greeted me, and I, as I often do in such situations, bluffed; he knew my name, but I couldn’t have told you who he was if my life depended on it.) Has my life-long pursuit of mathematical beauty hijacked parts of my brain intended to serve my social life and dragooned them into the service of mathematics? A sobering thought!
For more musings about Ramanujan and the neurological basis of doing mathematics, see Maria Isabel Garcia’s essay “We knew math before we knew words”.
RAMANUJAN’S DEATH AND AFTERLIFE
Ramanujan’s health declined during his last year in England, and he returned to India in 1919. At the time he was diagnosed with tuberculosis, but it’s now considered likely that he actually suffered from hepatic amoebiasis that, if diagnosed, could have been treated. After a year of illness, during which he continued to work on mathematics in the same high style as before, he died on April 26, 1920.
Ramanujan left us with an incomplete legacy. You may think I mean that he didn’t prove all his claims, but that’s not what I’m talking about. I’m talking about understanding, not proof. This isn’t to say that proof and understanding are disjoint — far from it. The best proofs don’t just certify that such and such a proposition is true; they also enlighten us as to why the proposition is true. Mathematicians often seek new proofs of already-proved propositions, because they desire not just the enhanced confidence that comes from independent corroboration but the richness of vision that comes from having multiple perspectives. And sometimes enlightenment doesn’t come from a proof in the standard sense; sometimes enlightenment comes from a story that answers the question “What ever led you to guess that such a thing might be true?” These are the stories that we are missing in the case of Ramanujan. Some of his formulas are like programs that have already been compiled; the actual source code, with all its helpful comment-lines, is missing. We can run the programs, and convince ourselves that they work, and maybe with strenuous effort prove that they work, but we can’t see why they work.
Almost as soon as the news of Ramanujan’s death reached England, mathematicians began the still-unfinished task of tending Ramanujan’s garden. When I was a graduate student, the leaders of this effort (or at least the ones I had contact with) were George Andrews and Bruce Berndt; they worked hard to reconstruct Ramanujan’s mathematical worldview, or in cases where that proved impossible, to construct alternative ways to arrive at Ramanujan’s conclusions. More recently, Ken Ono has been at the forefront of mathematicians continuing the development of Ramanujan’s ideas, and he has just written a memoir describing his own development as a mathematician and the impact Ramanujan had on his life. Check out this podcast and Ono’s book My search for Ramanujan: How I learned to count.
One reason I’m mentioning Ono is that, in addition to creating some great mathematics, he has asked a very important question: How can we make sure that potential Ramanujans in our own day and age are discovered and nurtured? A website called The Spirit of Ramanujan seeks to encourage young people from all over the world to solve challenging math problems and become part of the mathematical problem-solving community. I think we should support this effort, and especially applaud the fact that this is not a timed competition; quick thinkers aren’t always deep thinkers! For more on this project, and on Ono’s life, see John Pavlus’ article, listed at the end of this article.
But I’m not a hundred percent sure that Ramanujan himself would have tackled problems picked by others, no matter how well-chosen. Ramanujan wrote: “I have not trodden through the conventional regular course followed in a University course, but I am striking out a new path for myself.” There will always be people who choose to blaze new trails rather than widen existing ones; how do we recognize which of those people have the potential to revolutionize mathematics? That’s a harder question.
Thanks to Krishnaswami Alladi, George Andrews, John Baez, Bruce Berndt, Lawrence Cohen, Bill Gosper, Sandi Gubin, David Jacobi, Robert Kanigel, Evelyn Lamb, Ken Ono, “Shecky Riemann”, Steven Strogatz, James Tanton, and Allan Wechsler; if there are mistakes in this essay, (a) they are my sole responsibility, and (b) please let me know about them!
Also, apologies to Erich Segal for the opening sentence of my essay (though, from a legal perspective, parody means never having to say you’re sorry).
Next month (July 17): “The Man Who Knew Infinity”: what the film will teach you (and what it won’t).
#1: For all positive integers n, the number of compositions of n (that is, the number of ways of writing n as a sum of positive integers, where order matters) is 2n-1. (It’s amusing to see how much difference there is in this case between “counting where order does matter” and “counting where order doesn’t matter”; the former leads to an incredibly simple exact formula, while the latter gives us the p(n) sequence, which as we have seen required the combined efforts of Hardy and Ramanujan — and later efforts of Hans Rademacher — to tame with an exact formula.)
Why is this formula correct? Here an example supplemented by pictures may help. Let’s look at the four compositions of 3, namely 3, 2+1, 1+2, and 1+1+1. We can represent these four compositions by pictures:
Think of the circles as objects that we are arranging in piles, and the vertical lines as dividers. (So, for instance, the diagram for 2+4+1+1 would be o o|o o o o|o|o.) Given n circles, how many ways are there to insert dividers? There are n-1 spaces between dividers, and for each one, we can decide whether or not to put a divider there. Since we have n-1 binary choices, the number of possibilities is 2 times 2 times … times 2 (with n-1 factors of 2 in the product), or 2n-1.
Here’s a mathematical-historical nicety about how the formula of Hardy and Ramanujan compares with the formula of Rademacher: Hardy and Ramanujan’s formula, in its published form, was not a formula for p(n) in the strongest possible sense; it was an example of what is called an asymptotic expansion. If you tried to use the expansion to compute p(200), your approximations would get closer and closer for a while, and then annoyingly get farther and farther! That is, when k gets to be too much bigger than the square root of n, the kth Ramanujan approximation starts to give worse and worse estimates of p(n). (Using an asymptotic expansion is like writing the word “banana”: the trick is knowing when to stop!) Rademacher noticed that if you replace a certain expression in the formula (ex/2) by a different expression (sinh x), the annoying behavior goes away, and you get a good-old convergent expansion: as k goes to infinity, the absolute error goes to 0. Rademacher deserves credit for not just noticing this but proving it. But mathematician Atle Selberg pointed out that in Ramanujan’s original letters to Hardy one sees “sinh x“, not “ex/2″. So it’s reasonable to guess that if Hardy had digested Ramanujan’s ideas more deeply, or if Ramanujan had stuck with his original insight, they would have wound up with Rademacher’s series.
#2: Let’s use pictures of a different kind to prove Proposition A. Here are diagrams of the partitions of 4 containing at most 3 parts:
These diagrams represent the partitions 2+1+1, 3+1, 2+2, and 1+1+1+1, respectively (count the number of squares in each row of a diagram, going from top to bottom). If you flip each the four diagrams across its northwest-to-southeast diagonal, you get the four diagrams
Putting it more generally, if you take a diagram of this kind (called a Young diagram) that contains at most three rows (and therefore corresponds to a partition with at most three parts) and flip it across the diagonal, so that rows become columns and vice versa, you’ll get a Young diagram that contains at most three columns — that is, you’ll get a Young diagram in which each row is of length at most three, corresponding to a partition in which no parts exceed 3. Conversely, if you take a partition in which no parts exceed 3, encode it by a Young diagram, flip the Young diagram across the diagonal, and interpret the flipped graph as a partition, you’ll get a partition into at most three parts.
This is an example of what mathematicians call a bijection — a one-to-one correspondence between two sets (in this case, the set of all partitions of n with at most 3 parts and the set of all partitions of n with no parts exceeding 3). Admittedly, it might be a somewhat confusing bijection, because the two sets intersect one another! The reason mathematicians love bijections is that they give a principled way to see that two sets are the same size, without actually listing and counting all the elements. When you construct a bijection between two sets, you’re showing that the number of elements of the first set must equal the number of elements of the second set, even if you don’t know (yet) precisely what number that is!
#3: I’m not going to be able to walk you through the steps that might lead you to discover this solution to Proposition B on your own; if anyone knows of a good “motivated” explanation, please let me know!
Here’s a bijection that turns a partition into unequal parts (with no further restrictions) into a partition into odd parts (with no further restrictions): Write each term in the form m × 2k (where m is odd) and then replace the term m × 2k by a sum of 2k terms all equal to m.
Example: The partition 1+2+3+4+5+6 of the number 21, written as (1 × 1) + (1 × 2) + (3 × 1) + (1 × 4) + (5 × 1) + (3 × 2), becomes (1)+(1+1)+(3)+(1+1+1+1)+(5)+(3+3), or (in weakly increasing order) 1+1+1+1+1+1+1+3+3+3+5.
To see that this is really a bijection, we have to convince ourselves that the operation is reversible; e.g., starting from 1+1+1+1+1+1+1+3+3+3+5, how would we recover 1+2+3+4+5+6? Here we use the fact that every positive integer can be written in a unique way as a sum of unequal powers of 2. Write 1+1+1+1+1+1+1+3+3+3+5 as (1 × 7) + (3 × 3) + (5 × 1) (where the first number in each product is the odd number that’s being repeated, and the second number in each product is the number of times that the odd number occurs); then write the second number in each product as a sum of unequal powers of 2, obtaining 1 × (4+2+1) + 3 × (2+1) + 5 × (1); then apply the distributive law, obtaining (4+2+1) + (6+3) + (5); and then finally interpret what you’ve obtained as a new partition.
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