श्रीनिवास रामानुजन Srinivasa Ramanujan
Ideas
Counting partitions
A partition writes a number as a sum of positive whole numbers. Four has five partitions. The count grows very fast: 100 has 190,569,292. With Hardy he found an asymptotic formula for it. Along the way they developed the circle method. Hans Rademacher later turned it into an exact convergent series.
Heinz Klaus Strick: Srinivasa Ramanujan (MacTutor) · MacTutor: Ramanujan, Srinivasa (Dictionary of National Biography entry) · MacTutor History of Mathematics (St Andrews): Srinivasa Aiyangar Ramanujan · Jeffrey A. Harvey (University of Chicago): Ramanujan’s influence on string theory, black holes and moonshine (arXiv 1909.11477)
Results first and proofs later
He recorded about 3,900 results in his notebooks, mostly without proof. In his thinking intuition played a larger role than argument. Mathematicians have spent a century proving his claims. Most turned out true and a few false.
Wikipedia: Srinivasa Ramanujan · Encyclopedia.com: Ramanujan, Srinivasa Aaiyangar · Frank Aiello: Ramanujan’s Thoughts from God (arXiv 1707.03379) · Springer: Ramanujan’s Notebooks Part V by Bruce C. Berndt (1998)
Fast series for π
In 1914 he published series for 1/π that converge very fast. One term of one series gives about eight correct decimal places. IISc’s Aninda Sinha links them to the Chudnovsky algorithm used to compute π to 200 trillion digits. In 2025 IISc physicists found the same structure in logarithmic conformal field theories.
ThePrint: In Ramanujan’s 111-year-old formula IISc paper finds links with black holes and quantum theory (31 October 2025) · IISc: How Ramanujan’s formulae for pi connect to modern high energy physics (3 December 2025)
Mock theta functions
In his last letter to Hardy in January 1920 he described a new kind of function. He called them mock theta functions. The lost notebook holds most of his ideas on them. In 2012 physicists showed that the counting function for certain quantum black holes is a mock modular form.
Surya Teja Gavva (Rutgers University): The life of Ramanujan · Uncovering Ramanujan’s “Lost” Notebook: An Oral History (arXiv 1208.2694) · Wikipedia: Ramanujan’s lost notebook · Dabholkar, Murthy and Zagier: Quantum Black Holes, Wall Crossing and Mock Modular Forms (arXiv 1208.4074, 2012) · Jeffrey A. Harvey (University of Chicago): Ramanujan’s influence on string theory, black holes and moonshine (arXiv 1909.11477)
Taming divergent series
He assigned finite values to series that grow without limit. In a letter to Hardy he wrote that the sum of all whole numbers is minus one twelfth under his theory. String theory now uses the same value when it regularises zero-point energy.
the sum of an infinite no. of terms of the series: 1+2+3+4+⋯=-1/12 under my theory
Letter to G. H. Hardy (1913), as quoted by Jeffrey Harvey
Every number has a character
Hardy visited him in a nursing home in a taxi numbered 1729 and called it dull. Ramanujan said it was the smallest number that is a sum of two cubes in two different ways. 1729 is 1³ + 12³ and also 9³ + 10³. Numbers like it are now called taxicab numbers.
it is a very interesting number
Reply to Hardy, recalled in Hardy’s Royal Society obituary (1921)
G. H. Hardy: Obituary of Srinivasa Ramanujan, Proceedings of the Royal Society A (1921) · IMSc: D. A. B. Young on Ramanujan’s illness (after Notes and Records of the Royal Society, 1994) · Heinz Klaus Strick: Srinivasa Ramanujan (MacTutor)
Highly composite numbers
His Cambridge degree was for a study of numbers with more divisors than any smaller number, such as 24. Hardy said such numbers are as unlike a prime as a number can be. The work found subtle patterns in how divisors pile up.
MacTutor: Ramanujan, Srinivasa (Dictionary of National Biography entry) · ThePrint: Srinivasa Ramanujan, self-taught mathematician whose genius survives more than a century on (22 December 2020)
Mathematics and the divine
He was a practising Vaishnava and a strict vegetarian. His family revered the goddess Namagiri of Namakkal and he credited her with his gifts. His friends regarded him as something of a mystic. Hardy was an atheist and called the idea of divine dictation a fable.
An equation for me has no meaning unless it expresses a thought of God.
To a friend, as reported by his biographer S. R. Ranganathan
MacTutor: Ramanujan, Srinivasa (Dictionary of National Biography entry) · Krishnaswami Alladi (University of Florida): review of The Man Who Knew Infinity (Inference) · Hinduism Today: A Baffling Mind (October 2003)
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