A History of Mathematics From Mesopotamia to Modernity

Modernity and itsAnxieties 229

To take a ‘proper’ historical viewpoint one needs to have some understanding of what his work
meant to him, what it meant to number theorists in Cambridge at the time, and why it has con-
tinued to be important.^12 In the quotation above, he describes his interest in ‘divergent series’.
This at least restricts the field a little, and it helps to fix ideas (if it gives no idea of the scope of his
thinking) to remember that at the age of 17 he was investigating the simplest such series, 1 /n
(see Oresme, Chapter 6), and had calculated Euler’s constant

γ=nlim→∞

(

(^) kn= 1

1

k

−logn

)

to 15 decimal places. One sees an interest in infinity, and in its control; in regularity as infinity is

approached. And Hardy’s attempts to understand his friend’s thought—as between Cambridge pub-

lic school atheist and pious Brahmin—convey images of an infinity which is capable of generating

all primes.

Whatever ideas or traditions may have underlain Ramanujan’s thought, his practice was solidly

in a successful nineteenth-century tradition of hard number theory (modular functions) which

went back to Dirichlet. Indeed, his weakness in supplying proofs would not have been as suspect

in Dirichlet’s time as it was a hundred years later. However, he had the special advantages both of

working with Hardy (undoubtedly among the best in the field) and of his own ‘intuition’. Both of

these enabled him to go further than his contemporaries. It also helped that his directions were not,

in general, ones which anyone else had thought of or imagined would be worth pursuing. A famous

example, whose exact history still seems uncertain, is the formula for the partition number; and

since this (unlike some of his other work) is easy to describe, it is worth a mention.

For any natural numbern, the partition functionp(n)is defined to be the number of ways in

whichncan be written as a sum of natural numbers (unordered): easily,p( 1 )=1 ((1)),p( 2 )=

2 (( 2 ),(1, 1)),p( 3 )= 3 (( 3 ),(2, 1),(1, 1, 1)), andp( 4 )=5 ((4), (3,1), (2,2), (2,1,1), (1,1,1,1)

in obvious notation). Work had been done since the time of Euler to find formulae forp(n); and

Ramanujan claimed that he had such an exact formula. Hardy was unable to believe this—the usual

best hope would be for an ‘asymptotic formula’, which describes limiting behaviour. Naturally, too,

Ramanujan was unable to explain, if he knew, why his formula was right. As a result, what we

now have is known as the rigorously proved ‘Hardy–Ramanujan asymptotic formula’, found in

1916–17:

p(n)∼

1

4 n

√

3

eπ

√ 2 n/ 3

Here ‘an∼bn’ means thatan/bn→1asn→∞. The formula is at least relatively simple, and

gives an idea of how fast the partition function grows. The much more complicated exact function,

which may or may not be what Ramanujan found, was proved by Rademacher in 1937.

To say that Ramanujan’s mathematics was tangential, let alone marginal to twentieth-century

mathematics would be absurd given the influence of his published work, his unproved conjectures,

let alone his unedited notebooks. Perhaps his work should stand rather as an image of modern

mathematics’ capacity for absorbinganythinghowever ‘different’ and setting it to work in its vast

theorem factory.

12. To the extent that (for example) Deligne’s solution of the Ramanujan conjecture in the 1970s with the full apparatus of late
    twentieth-century algebraic geometry (I forbear to give details) is for many mathematicians an achievement on a level with Wiles’s
    proof of the Taniyama–Shimura conjecture.