Explorers of Infinity

A relationship that turned the world of mathematics upside down

“The Riemann Hypothesis was born out of an encounter, …a great fusion, between counting logic and measuring logic. To put it in precise mathematical terms; it arose when some ideas from arithmetic were combined with some from analysis to form a new thing, a new branch of the mathematical tree, analytic number theory.”

• An excerpt from ‘Prime Obsession’

For a mathematically curious person, ‘Prime Obsession’ is a mind stretching journey to one of the most profound mysteries in mathematics — the Riemann Hypotheses. First published in Riemann’s groundbreaking 1859 paper, the generalized hypothesis conjectures that

All non-trivial zeros of the zeta function have real part one-half.

Euler’s zeta function was extended by Riemann to the entire complex plane, while studying the distribution of prime numbers.

He noted that all the non-trivial zeta function zeros -2, -4, -6, … would therefore lie on the “critical line”.

One of the last great unsolved problems in Math, it has thus far resisted all attempts to prove it. Even the combined mathematical prowess of Hardy and Ramanujan could not achieve this feat. (Riemann Hypotheses has been widely blamed for Ramanujan’s and Hardy’s instabilities, hence giving it somewhat a reputation of being a curse)

Reading more about this theorem and the world of prime numbers, I somehow ended up going through some of the most famous works of the brilliant English Mathematician G.H. Hardy.

With carefully worded, accurate proofs ‘Orders of Infinity’ — a mathematical pamphlet written by Hardy, he had recast theorems dealing with limit-behavior of functions (obtained by Du Bois-Reymond in a series of papers dating from 1871 to 1880) according to the then requirements.

And his paper might well have given rise to the beginning of the much-celebrated collaboration between Ramanujan & Hardy.

An excerpt from the first letter Ramanujan wrote to Hardy –

‘Dear Sir,

I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only £20 per annum. I am now about 23 years of age. I have had no University education, but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course, which is followed in a University course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as ‘startling.’

An insignificant clerk in an Indian office at that time, Ramanujan in his letter went on to insist that he could give meaning to negative gamma functions and also disputed an assertion made by Hardy in his ‘Orders of Infinity’ paper. Most of all, he declared he had found ways to better the famous ‘prime number theory’. Enclosing a part of his findings along with the letter, he signed off asking for Hardy’s advice on whether he should go ahead and publish his theorems.

Robert Kanigel, in his biography on the self-taught Indian mathematician ‘Srinivasa Ramanujan’, remarks that ‘For Hardy, Ramanujan’s pages of theorems were like an alien forest whose trees were familiar enough to call trees, yet so strange they seemed to have come from another planet.’

The strangeness of Ramanujan’s theorems was what struck him first, not his brilliance.

Hardy put off the whole episode labeling Ramanujan as just another crank. But for the remainder of that day, the wild theorems he had seen in the morning kept repeating in his head. And by dinner time, he had decided to call up on his colleague, Littlewood, to further examine the papers.

Hours of rummaging through the stuff and the duo were finally beginning to realize that they were indeed looking at papers of a mathematical genius.

Ramanujan’s religious theology forbade him from crossing the seas, but Hardy was persistent and somehow convinced the man to move to Cambridge. Under Hardy’s mentorship, Ramanujan created, proved, negated mathematical theorems that were seldom thought of before. In a memorial speech at Harvard in 1936, Hardy said,

“They [Ramanujan’s theorems] must be true, because if they were not true, no one would have the imagination to invent them.”

It has been estimated that Ramanujan conjectured/proved close to 3000 theorems, equations, identities, and properties of highly composite function, in his lifetime. Not forgetting major investigations, he carried out in areas of mock theta functions, gamma functions, divergent series and prime number theory.

Not being able to withstand the climate and dietary conditions of England, Ramanujan returned to India in 1919, after the war got over. But soon there was a relapse in his illness, and he was diagnosed with tuberculosis.

Around January 1920, Ramanujan wrote to Hardy for the first time in almost a year. He explained his discovery and subsequent work on the ‘Mock Theta’ functions. Robert Kanigel terms this work as Ramanujan’s ‘Final Problem’ (refer to Sir Arthur Conan Doyle’s story by the same name).

During his final year, with an approaching death and an absolute physical deterioration, Ramanujan worked relentlessly on his discovery. He filled page after page with theorems and computations clocking almost 650 formulae in total.

Through all the sufferings and household squabbles, Ramanujan till the very end of his life, lay on his bed with his head propped up on pillows — working. As his wife, Janaki, remarked — ‘It was always maths.’

On April 26, 1920, Ramanujan passed away at a tragic age of 32. His remaining handwritten notes, consisting of formulae on singular moduli, hypergeometric series, and continued fractions, were compiled by his brother, Tirunarayanan.