One Sunday morning soon after the December issue appeared, P. C. Mahalanobis sat with it at a table in Ramanujan’s rooms in Whewell’s Court. Mahalanobis was the King’s College student, just then preparing for the natural sciences Tripos, who had found Ramanujan shivering by the fireplace and schooled him in the nuances of the English blanket. Now, with Ramanujan in the little back room stirring vegetables over the gas fire, Mahalanobis grew intrigued by the problem and figured he’d try it out on his friend.
“Now here’s a problem for you,” he yelled into the next room
“What problem? Tell me,” said Ramanujan, still stirring. And Mahalanobis read it to him.
“I was talking the other day,” said William Rogers to the other villagers gathered around the inn fire, “to a gentleman about the place called Louvain, what the Germans have burnt down. He said he knowed it well—used to visit a Belgian friend there. He said the house of his friend was in a long street, numbered on this side one, two, three, and so on, and that all the numbers on one side of him added up exactly the same as all the numbers on the other side of him. Funny thing that! He said he knew there was more than fifty houses on that side of the street, but not so many as five hundred. I made mention of the matter to our parson, and he took a pencil and worked out the number of the house where the Belgian lived. I don’t know how he done it.”
Perhaps the reader may like to discover the number of that house.
Through trial and error, Mahalanobis (who would go on to found the Indian Statistical Institute and become a Fellow of the Royal Society) had figured it out in a few minutes. Ramanujan figured it out, too, but with a twist. “Please take down the solution,” he said—and proceeded to dictate a continued fraction, a fraction whose denominator consists of a number plus a fraction, that fraction’s denominator consisting of a number plus a fraction, ad infinitum. This wasn’t just the solution to the problem, it was the solution to the whole class of problems implicit in the puzzle. As stated, the problem had but one solution—house no. 204 in a street of 288 houses; 1 + 2 + … 203 = 205 + 206 + … 288. But without the 50-to-500 house constraint, there were other solutions. For example, on an eight-house street, no. 6 would be the answer: 1 + 2 + 3 + 4 + 5 on its left equaled 7 + 8 on its right. Ramanujan’s continued fraction comprised within a single expression all the correct answers.
Mahalonobis was astounded. How, he asked Ramanujan, had he done it?
“Immediately I heard the problem it was clear that the solution should obviously be a continued fraction; I then thought, Which continued fraction? And the answer came to my mind.”
4. THE ZEROES OF THE ZETA FUNCTION
The answer came to my mind. That was the glory of Ramanujan—that so much came to him so readily, whether through the divine offices of the goddess Namagiri, as he sometimes said, or through what Westerners might ascribe, with equal imprecision, to “intuition.” And yet, it was the very power of his intuition that, in one sense, undermined his mathematical development. For it blinded him to intuition’s limits, gave him less reason to learn modern mathematical tools, shielded him from his own ignorance.
“The limitations of his knowledge were as startling as its profundity,” Hardy would write.
Here was a man who could work out modular equations and theorems of complex multiplication, to orders unheard of, whose mastery of continued fractions was, on the formal side at any rate, beyond that of any mathematician in the world, who had found for himself the functional equation of the Zeta-function, and the dominant terms of many of the most famous problems in the analytic theory of numbers; and he had never heard of a doubly periodic function or of Cauchy’s theorem, and had indeed but the vaguest idea of what a function of a complex variable was. His ideas as to what constituted a mathematical proof were of the most shadowy description. All his results, new or old, right or wrong, had been arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account.
That mysterious “process” sometimes led him seriously astray. And Hardy’s reference to the zeta function embodied one notorious example of it.
In Ramanujan’s first letter, he’d referred to a statement in Hardy’s Orders of Infinity concerning prime numbers. He had, he wrote, “found a function which exactly represents the number of prime numbers less than x,” in the form of an infinite series. Very interesting, Hardy had written back, let’s see some proof. In Ramanujan’s next letter, he elaborated. “The stuff about primes is wrong,” Littlewood shot back to Hardy when he saw it. And now, with Ramanujan in England, Hardy came to see, up close, just how he had gone wrong.
Ramanujan was not the first mathematician to be bewitched by prime numbers. The primes—numbers like 2, 3, 5, 7, and 11, but not 6 or 9 (respectively, 2 × 3 and 3 × 3, and so “composite”)—were the building blocks of the number system. Start counting and it was hard to help wondering, when would you get to a prime? Was there any sort of pattern? If you laid out all the numbers on a giant grid, colored in the primes and left the others blank, then stood back and took a picture of the resulting pattern, would there be anything to see, any pattern you could call a pattern?
At first glance there wasn’t, certainly nothing obvious. And yet, then again, there was something there, right in front of your eyes: as you kept on counting, you found more primes. There was no last prime. Euclid had proved as much twenty-three hundred years ago. “The primes are the raw material out of which we have to build arithmetic,” Hardy would write, “and Euclid’s theorem assures us that we have plenty of material for the task”; just as you never run out of numbers, you never run out of primes.
Well, then, that was something. But could you say anything more? Yes: as you kept counting it looked, on average, as if you encountered a lower density of primes. There were always more of them, but the rate at which you encountered them dropped off. In the first one hundred numbers, for example, there were twenty-five primes; in the second hundred, twenty-one; in the ninth hundred, fifteen. Occasionally, you ran into a span of higher density; there are sixteen primes between 1100 and 1200, for example. But on average the falloff held.
Anything else? Yes, again. While the density of primes fell off, it fell off slowly. Deliberately, inexorably, the slowing effect could be felt; but it took hold oh-so-gradually. This was not like a campfire roaring away at dusk that is nothing but embers by midnight, and cold by morning; but rather one that by sunrise still burned brightly, only not quite so brightly. With primes, by the time you were in the billions, you still got five of them in every 100 numbers, compared to fifteen per hundred in the first thousand. So the braking effect was a gentle one.
Could you be more precise than that? Could you say just how slow the slowing was? Could you, to put it another way, make a mathematical statement that gave the number of primes you encountered in counting to any given number?
In time, mathematicians guessed that the restraining influence, the mathematical “force” slowing the increase in the number of primes, was logarithmic; that the mathematical function known as the logarithm was somehow at work.
To say that something “drops off logarithmically” is the reverse of saying that it “increases exponentially.” Exponential growth implies a kind of takeoff, rising and re-rising, ever more rapidly, on the strength of its own growth; compound interest, the kind that yields those thick retirement larders thirty years down the road, is an exponential process, rising slowly at first, then accelerating. But with logarithmic growth, quite the opposite occurs: you get less and less bang for the buck.
The logarithmic response is ubiquitous in nature, as, for example, in the realm of the human senses. Double the amount of light in a room and you scarcely notice; the eye can respond to both the glare of the noonday sun and to the flicker of a match a mile away because its response is logarithmic. If light intensity rises by a factor of a thousand, which can be written as 103, the response, in principle, may be only about three times as mu
ch instead of a thousand.
Well, it was just such a slow logarithmic growth in this braking effect that mathematicians long thought they saw with prime numbers. Or, mathematically
Here π(x), read “pi of x,” means the number of primes encountered in the first x numbers. And this, the equation says, is equal to simply x divided by the (“natural”) logarithm of x. In fact, this is not quite right. Rather,
This expresses the same idea except as an approximation, thought to get better and better as x grows larger. (Gauss actually wrote it as a logarithmic integral, which is similar in principle.)
In 1896, the French mathematician Hadamard and the Belgian de la Vallée-Poussin proved the long-conjectured prime number theorem, as this relationship was known. And that’s where things stood in 1914: As x grew large, the theorem said, a key ratio—π(x) ln x/x—approached one. That was pretty good, certainly putting things on a firm mathematical basis. And yet, it was still an approximation, one that gave no clue to its accuracy; it failed to specify how “fast” the ratio approached one, much less give you a formula that said, give me the number to which you counted, and I’ll tell you how many primes you’ve passed along the way. It couldn’t say, for example, that among the first million integers there are, exactly, 78,498 primes. And it was this that Ramanujan in his letters to Hardy confidently declared he could do: I have found a function which exactly represents the number of prime numbers less than x.
Well, he hadn’t.
Ramanujan’s formula, three versions of which he gave in his second letter, was an infinite series which, for values of x up to 1000 for example, gave virtually exact agreement. Even larger values, Hardy found later, were strikingly close. Of the first nine million numbers, it was known that 602,489 were prime. Ramanujan’s formula gave a figure off by just 53—closer than the canonical version of the prime number theorem. What was more—and this was the crucial point—any error accompanying its use, he claimed, was bounded, stayed within given limits. This was why, in his first letter, he’d objected to Hardy’s assertion that the “precise order” of the error term “has not yet been determined”; his formulas, he insisted, did determine it.
But they didn’t. Back in 1913, shortly after Hardy’s first encouraging letter had arrived, Ramanujan’s friend Narayana Iyer had slipped into the Journal of the Indian Mathematical Society some of Ramanujan’s results on primes. “Proofs,” he added, “will be supplied later.” But they never were. Because they didn’t exist—couldn’t exist. Because Ramanujan’s conclusions were wrong.
Once Ramanujan was in England, Hardy studied Ramanujan’s notebooks, listened as he worked through his arguments—and came to see where he had stumbled. Ramanujan had been misled by undue reliance on the low values of x for which he’d tried his formula; the error, for higher values of x, was much larger than he thought. These, had he tried them, might have alerted him to the more basic flaws in his approach, which Hardy would find so illuminating that one day he’d build a whole lecture around them. Ramanujan’s theory, Hardy would write, “was (so to say) what the theory might be if the zeta function had no complex zeroes.”
The Riemann zeta function was a simple enough looking infinite series expressed in terms of a complex variable. Here, “complex” means not difficult or complicated, but refers to a variable of two distinct components, “real” and “imaginary,” which together could be thought to range over a two-dimensional plane. In 1860, Georg Friedrich Bernhard Riemann made six conjectures concerning the zeta function. By Ramanujan’s time, five had been proven. One, enshrined today as the Riemann hypothesis, had not.
If you set the zeta function equal to zero, Riemann conjectured, then certain solutions to the resulting equation—its “complex zeroes”—would, when graphed, all lie along a particular line, one parallel to the “imaginary” axis and half a unit to its right. And from this hypothesis, if valid, certain important conclusions about the distribution of primes, going beyond the prime number theorem itself, would automatically flow.
But was the Riemann hypothesis true? To this day, no one knows, and it remains one of the great unproved conjectures in mathematics. Around the time Ramanujan arrived in England, Hardy proved something just short of it—that an infinite number of solutions lay on the crucial line. But this was not the same as saying they all did. The renowned German mathematician David Hilbert once said that, were he awakened after having slept for a thousand years, his first question would be, Has the Riemann hypothesis been proved?
Ramanujan had come up with something like Riemann’s zeta function on his own but had misunderstood what he’d found. He had ignored the crucial complex zeroes, acted as if they didn’t exist—leading to a version of the prime number theorem that was, simply, wrong. “There are regions of mathematics in which the precepts of modern rigour may be disregarded with comparative safety,” Hardy would write, “but the Analytic Theory of Numbers is not one of them.”
As Littlewood would write of Ramanujan’s effort on primes, “These problems tax the last resources of analysis, took over a hundred years to solve, and were not solved at all before 1890 [sic; he probably meant 1896, when the prime number theorem was proved]; Ramanujan could not possibly have achieved complete success. What he did was to perceive that an attack on the problems could at least be begun on the formal side, and to reach a point at which the main results become plausible. The formulae do not in the least lie on the surface, and his achievement, taken as a whole, is most extraordinary.”
Hardy would be similarly indulgent of Ramanujan’s errors. “I am not sure,” he would write, “that, in some ways, his failure was not more wonderful than any of his triumphs.” He later regretted saying that, dismissing it as unduly sentimental. But he was getting at something—that in reaching his faulty conclusions, Ramanujan had rediscovered the prime number theorem; and that, in trying to reach beyond it, he had pursued arguments that, though technically flawed, were, in their own way, brilliant.
Still, Ramanujan was wrong, which now, in England, under Hardy’s tutelage, he came to understand. “His instincts misled him,” wrote Hardy. And that was the point. Ramanujan’s “instincts,” sure as they were, in some ways better than those of any other mathematician of his day, were not good enough.
• • •
A car mechanic reliant on mechanical instinct may “know” how an engine works yet be unable to set down the physical and chemical principles governing it. For a writer, it may be enough to “know” that one scene should precede another and not follow it, without being able to explain why. But mathematicians are not normally content to guess, or assume, or assert that something is true; they must prove it, or feel they have—or as Hardy would put it, “exhibit the conclusion as the climax of a conventional pattern of propositions, a sequence of propositions whose truth is admitted and which are arranged in accordance with rules.”
Proof is no mere icing on the cake. Take the sequence of integers 31, 331, 3331, 33331, 333331, 3333331. Each is a prime number. So is the next in the sequence. Have we hit upon some hidden pattern? No, the pattern self-destructs with the next in line, the product of 17 and 19,607,843. Or what about numbers of the form 22n + 1. For n = 0, 1, 2, 3, and 4, the resulting numbers are all prime. Are they for all n? Pierre de Fermat conjectured as much. But he conjectured wrong. Because, as Euler found, even the next number is not prime, but the product of 641 and 6,700,417.
Many other examples like this, where seemingly “obvious” patterns prove not to be patterns at all, appear all through number theory and elsewhere in mathematics. One Hardy liked to cite also emerged from the theory of primes. Over the years, in comparing the approximations of the prime number theorem with the actual number of primes calculation revealed, the approximation always proved higher; the error was always in the same direction. You could try a thousand, you could try a million, you could try a billion, you could try a trillion, you could try a billion trillion, and it always came out the same, making all the forces of intui
tion argue that it was always so—and that a theorem embodying it would be a great one to run off and prove. But no such theorem could ever be proved. Because, intuitively obvious though it might seem, it simply wasn’t true.
A year before he heard from Ramanujan, Littlewood had proved that, if you went far enough, the prime number theorem was destined to sometimes predict less, not more, than the actual number of primes. Later, someone found the number below which this reversal was guaranteed to take place. And it was a number so big you had to laugh—a number more than the number of particles in the universe, more than the possible games of chess. It was, Hardy would say, “the largest number which has ever served any definite purpose in mathematics.” And it made for the ultimate illustration of how intuition could serve you badly, and so must always be subject to proof.
To “prove” something, then, is a kind of guarantee—that, for mathematical entities A, B, and C, and subject to constraints D, E, and F, the theorem holds. A theorem must stand up to hard use, will often be applied to unanticipated new situations. Fail to precisely fix the conditions under which it applies, and you’re apt to go wrong. A mathematician with an insufficiently ironclad proof is a little like the brash young police lieutenant in the movies convinced of the butler’s guilt but brought up short by his boss’s caution, Yeah, but that won’t convince a jury.
Throughout his notebooks, all during his time in India, and all through his early letters, Ramanujan had proclaimed a thousand versions of the butler did it. Most of the time he was right, and the butler had done it: his results were true. And yet before coming to England, he would have been unable to secure a conviction: he had—to extend the metaphor—scant knowledge of the rules of evidence, or the relevant criminal law, or even the standards of sound legal argument. Ramanujan’s “proof” of his new version of the prime number theorem was no proof at all.
The Man Who Knew Infinity Page 28