Ramanujan: The Self-Taught Genius Who Changed Number Theory

From Erode to Kumbakonam

Srinivasa Ramanujan was born on 22 December 1887 in Erode, a small town in the Madras Presidency (now Tamil Nadu) of British India. His family was Brahmin but poor. His father, K. Srinivasa Iyengar, worked as a clerk in a cloth merchant's shop, and his mother, Komalatammal, was a deeply religious woman who sang at a local temple.

Ramanujan grew up in Kumbakonam, a town known for its temples and its academic tradition. He showed an extraordinary affinity for mathematics from an early age. By the time he was eleven, he had exhausted the mathematical knowledge of two college students who lodged at his family's home. By thirteen, he had mastered S.L. Loney's Trigonometry, a book intended for university students.

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The Book That Changed Everything

At the age of fifteen, Ramanujan obtained a copy of George Shoobridge Carr's A Synopsis of Elementary Results in Pure and Applied Mathematics, a compilation of approximately 5,000 theorems and formulas — mostly stated without proof. This book became Ramanujan's primary mathematical companion. Working through it, he not only verified the results but began generating his own, filling notebooks with original formulas, identities, and conjectures.

The style of Carr's book — results without proofs — profoundly shaped Ramanujan's own style. Throughout his life, he would produce formulas of stunning beauty and depth, often without anything resembling a conventional proof.

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Years of Struggle

Ramanujan's obsessive focus on mathematics came at a cost. He won a scholarship to the Government Arts College in Kumbakonam in 1904 but lost it because he neglected all subjects other than mathematics. He tried again at Pachaiyappa's College in Madras but again failed his non-mathematical examinations.

For several years, he lived in poverty, often on the edge of starvation. He worked as a clerk at the Madras Port Trust, earning a modest salary, while filling notebook after notebook with mathematical discoveries. He had no formal mathematical training beyond what he had taught himself.

During this period, Ramanujan approached several Indian mathematicians for support. Some recognized his talent, particularly Ramachandra Rao, who provided financial support, and V. Ramaswamy Aiyer, who helped him publish his first paper in the Journal of the Indian Mathematical Society in 1911.

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The Letter to Hardy

In January 1913, Ramanujan wrote a letter to G.H. Hardy, a leading mathematician at the University of Cambridge. The letter contained a selection of approximately 120 formulas and theorems, presented without proof. Hardy later recalled his reaction:

Among the results in the letter were remarkable formulas such as Ramanujan's rapidly converging series for $\frac{1}{\pi}$:

This series converges so rapidly that each additional term adds approximately eight correct decimal digits of $\pi$. It was used as the basis for some of the fastest algorithms for computing $\pi$ in the late twentieth century.

Hardy, together with his collaborator J.E. Littlewood, studied the letter carefully and concluded that Ramanujan was a mathematician of extraordinary originality. Hardy arranged for Ramanujan to come to Cambridge.

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Cambridge: 1914–1919

Ramanujan arrived in England in April 1914, just months before the outbreak of World War I. The transition was difficult — he was a strict vegetarian in a country at war, far from family, and struggling with the cold English climate.

Mathematically, however, the Cambridge years were extraordinarily productive. Working with Hardy, Ramanujan published numerous important papers and developed a formal framework for many of his earlier discoveries.

The Partition Function

One of the most celebrated results of the Hardy–Ramanujan collaboration concerns the partition function $p(n)$, which counts the number of ways to write a positive integer $n$ as a sum of positive integers. For example, $p(4) = 5$ because:

$4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1$

Hardy and Ramanujan developed an asymptotic formula for $p(n)$:

$p(n) \sim \frac{1}{4n\sqrt{3}} \exp\left(\pi\sqrt{\frac{2n}{3}}\right) \quad \text{as } n \to \infty$

Their method — the circle method — became one of the most powerful tools in analytic number theory. It was later refined by Hardy and Littlewood, and then by Rademacher, who obtained an exact convergent series for $p(n)$.

Ramanujan's Congruences

Ramanujan discovered remarkable congruence properties of the partition function:

$p(5n + 4) \equiv 0 \pmod{5}$ $p(7n + 5) \equiv 0 \pmod{7}$ $p(11n + 6) \equiv 0 \pmod{11}$

These congruences hinted at deep structural properties of partitions that were not fully understood until the work of Ken Ono and others many decades later, using the theory of modular forms.

Highly Composite Numbers

Ramanujan's 1915 paper on highly composite numbers — positive integers with more divisors than any smaller positive integer — was over 50 pages long and was his longest published work. The sequence begins:

$1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, \ldots$

He developed a complete theory of their structure, showing that they have the form $2^{a_1} \cdot 3^{a_2} \cdot 5^{a_3} \cdots$ where $a_1 \geq a_2 \geq a_3 \geq \cdots$.

The Ramanujan Tau Function

Ramanujan studied the function $\tau(n)$ defined by the expansion of the discriminant modular form:

$\Delta(q) = q \prod_{n=1}^{\infty} (1 - q^n)^{24} = \sum_{n=1}^{\infty} \tau(n) q^n$

He conjectured that $\tau$ is multiplicative (i.e., $\tau(mn) = \tau(m)\tau(n)$ when $\gcd(m,n) = 1$) and that $|\tau(p)| \leq 2p^{11/2}$ for all primes $p$. The multiplicativity was proved by Mordell in 1917; the bound — the famous Ramanujan conjecture — was proved by Deligne in 1974 as a consequence of his proof of the Weil conjectures.

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The Notebooks

Ramanujan left behind three notebooks and a collection of loose papers (the "lost notebook," rediscovered in 1976 by George Andrews in the Trinity College Library). These notebooks contain approximately 3,900 results — formulas, identities, and conjectures — many of which were decades ahead of their time.

Highlights include:

• Mock theta functions, which Ramanujan described in his last letter to Hardy in 1920. Their proper mathematical framework was not understood until Sander Zwegers's 2002 thesis and the subsequent work of Kathrin Bringmann and Ken Ono.
• Rogers–Ramanujan identities:

$\sum_{n=0}^{\infty} \frac{q^{n^2}}{(1-q)(1-q^2)\cdots(1-q^n)} = \prod_{n=0}^{\infty} \frac{1}{(1-q^{5n+1})(1-q^{5n+4})}$

These identities connect number theory, combinatorics, and statistical mechanics.

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The Taxicab Number

A famous anecdote captures Ramanujan's extraordinary intuition. Hardy once visited Ramanujan in the hospital and mentioned that he had arrived in taxi number 1729, which seemed to him a rather dull number. Ramanujan immediately replied:

Indeed, $1729 = 1^3 + 12^3 = 9^3 + 10^3$. Numbers with this property are now called taxicab numbers in his honor.

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Illness and Death

The English climate and wartime food shortages took a severe toll on Ramanujan's health. He was diagnosed with tuberculosis (recent scholarship suggests it may have been hepatic amoebiasis contracted in England). He returned to India in 1919, gravely ill.

Ramanujan continued to work on mathematics from his sickbed, producing some of his most profound results, including the mock theta functions. He died on 26 April 1920, at the age of 32, in Kumbakonam.

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Legacy

Ramanujan's influence on mathematics has only grown with time. His work on modular forms, $q$-series, and mock theta functions anticipated developments that came decades later. The Ramanujan conjecture inspired the Langlands program, one of the deepest organizing principles in modern mathematics.

Every year on 22 December — Ramanujan's birthday — India celebrates National Mathematics Day. The Ramanujan Prize, awarded annually by the International Centre for Theoretical Physics, honors young mathematicians from developing countries.

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References

• Hardy, G.H., Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, Cambridge University Press, 1940.
• Kanigel, R., The Man Who Knew Infinity: A Life of the Genius Ramanujan, Charles Scribner's Sons, 1991.
• Berndt, B.C., Ramanujan's Notebooks, Parts I–V, Springer, 1985–1998.
• Andrews, G.E. and Berndt, B.C., Ramanujan's Lost Notebook, Parts I–IV, Springer, 2005–2013.
• Wikipedia — Srinivasa Ramanujan
• MacTutor — Srinivasa Ramanujan
• The Ramanujan Journal — Springer