Ramanujan & Hardy

21.1 G.H. Hardy (1877–1947)

Godfrey Harold Hardy was born on 7 February 1877 in Cranleigh, Surrey, into a family of teachers. Both his parents were mathematically gifted, and young Harold showed extraordinary numerical facility from infancy. He attended Winchester College and then Trinity College, Cambridge, where he became a Fellow in 1900. He remained at Cambridge (with a period at Oxford from 1919 to 1931) for the rest of his career, becoming one of the most influential pure mathematicians of the twentieth century.

Hardy was a passionate advocate for pure mathematics — mathematics pursued for its own beauty, without regard to practical applications. His 1940 essay A Mathematician’s Apology is one of the most eloquent and famous defenses of pure mathematics ever written. In it he argued that the best mathematics is that which has no practical application, and that mathematics is ultimately justified by its aesthetic value, not its utility. Ironically, some of his work in number theory later found applications in cryptography, and the Hardy–Weinberg principle became a cornerstone of population genetics.

Hardy’s mathematical contributions were vast. He made fundamental advances in analytic number theory (with Littlewood), the theory of series and integrals, Diophantine approximation, and the Riemann zeta function. His collaboration with John Edensor Littlewood(beginning around 1911 and lasting 35 years) was one of the most productive partnerships in the history of mathematics, producing about 100 joint papers. The Hardy–Littlewood conjectures on the distribution of prime numbers remain influential and largely open.

Hardy was also known for his mathematical eccentricities. He had a famous list in which he rated mathematicians on a scale of talent: he gave himself 25, Littlewood 30, Hilbert 80, and Ramanujan 100. He despised applied mathematics, once toasting: “Here’s to pure mathematics! May it never be of use to anyone!” He was also an avid cricket fan who used to analyze batting averages with mathematical precision.

The Hardy–Littlewood Circle Method

One of Hardy’s most important technical contributions was the development of the circle method (also called the Hardy–Littlewood method), originally conceived in collaboration with Ramanujan and then refined extensively with Littlewood. The basic idea is to represent a counting function as a contour integral around the unit circle, then decompose the circle into “major arcs” (near rational numbers with small denominators, which contribute the main term) and “minor arcs” (the remainder, which must be shown to be negligible).

In schematic form, if we want to count the number of representations $r(n)$ of $n$ in a certain way, we write:

The circle method was first used to study the partition function (yielding the Hardy–Ramanujan asymptotic formula) and was then applied to Waring’s problem, Goldbach’s conjecture, and many other problems in additive number theory. It remains one of the most powerful techniques in the field and has been extended by Vinogradov, Davenport, and many others. The Hardy–Littlewood conjecture on prime k-tuples, formulated using heuristic reasoning from the circle method, continues to guide research in prime number theory to this day.

21.2 Srinivasa Ramanujan: Childhood and Self-Education

Srinivasa Ramanujan Iyengar was born on 22 December 1887 in Erode, in the Tamil Nadu region of southern India, into a poor Brahmin family. His father, K. Srinivasa Iyengar, was a clerk in a sari shop earning a meager salary; his mother Komalatammal was a devout woman who sang devotional songs at the local Namagiri temple. The family lived in the town of Kumbakonam, known as the “Cambridge of South India” for its many temples and schools.

From early childhood, Ramanujan was obsessed with numbers and mathematical patterns. By age 11, he had exhausted the mathematical knowledge of two college students who boarded at his home. By 12, he had mastered advanced trigonometry, including results that he discovered independently. He derived Euler’s formula for the relationship between trigonometric functions and the exponential function on his own, unaware that Euler had discovered it more than a century earlier. When he later learned this fact, he was reportedly so disappointed that he hid his notebook in the roof of his house.

At age 15, he obtained a copy of G.S. Carr’s A Synopsis of Elementary Results in Pure and Applied Mathematics (1886), a compendium of about 5,000 formulas presented without proofs. This unusual book became Ramanujan’s primary mathematical education. Carr’s style — bare statements of results with minimal explanation — profoundly shaped Ramanujan’s own approach. He worked through each formula, supplying his own proofs and discovering vast generalizations. The book effectively ended at roughly the 1860s, meaning Ramanujan was largely unaware of developments in the previous half-century of mathematics.

Ramanujan’s formal education was patchy and ultimately unsuccessful in conventional terms. In 1904, he entered the Government Arts College in Kumbakonam on a scholarship, but he was so consumed by mathematics that he failed his other subjects and lost the scholarship. He tried again at Pachaiyappa’s College in Madras in 1906, but again failed in English, Sanskrit, and physiology. He never obtained a university degree through regular coursework.

During these years of academic failure, Ramanujan lived in near-poverty, sometimes going hungry, supported occasionally by patrons who recognized his talent. He wandered the streets of Madras with his notebooks, showing his results to anyone who might listen. His mathematical work during this period was recorded in three thick notebooks that would later prove to be treasure troves of new results. He filled these notebooks with thousands of formulas on infinite series, continued fractions, number theory, definite integrals, and modular equations — an entire private mathematical universe constructed in isolation.

21.3 The Famous Letter to Hardy (1913)

On a January morning in 1913, G.H. Hardy received a bulky envelope postmarked Madras, India. Inside was a letter from an unknown 25-year-old clerk named S. Ramanujan, working at the Madras Port Trust on a salary of roughly £20 per year. The letter began:

“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.”

The letter then plunged into page after page of extraordinary mathematical formulas — about 120 theorems in all, presented without proofs. Hardy initially suspected a hoax, then spent the evening studying the formulas with Littlewood. Some were well-known results; others were new to him. Some he could prove; others defeated him completely. As he later recalled:

“They must be true, because, if they were not true, no one would have had the imagination to invent them.”

Among the specific formulas in the letter were remarkable results. One class involved rapidly converging infinite series:

Hardy recognized that Ramanujan was a mathematical genius of the first order — a “natural genius” in the mold of Euler and Jacobi. He immediately began arrangements to bring Ramanujan to Cambridge, a process that took over a year due to Ramanujan’s Brahmin religious restrictions against overseas travel, his mother’s initial opposition, and bureaucratic obstacles. Ramanujan’s mother finally relented after reportedly receiving a vision from the goddess Namagiri granting permission.

Hardy later described the discovery of Ramanujan as “the one romantic incident in my life.”It was also, by his own assessment, his greatest contribution to mathematics — not any theorem he proved himself, but the recognition and nurturing of Ramanujan’s extraordinary talent. Hardy noted that Ramanujan had written to two other Cambridge mathematicians before him (H.F. Baker and E.W. Hobson), both of whom returned the letter without comment.

21.4 The Cambridge Years (1914–1919)

In April 1914, Ramanujan sailed for England. The five years in Cambridge were the most productive of his life, but also the most difficult. He struggled with the English climate, could not find vegetarian food suitable to his Brahmin dietary requirements (made worse by wartime rationing), suffered from loneliness and depression, and eventually contracted tuberculosis (or possibly hepatic amoebiasis, as more recent medical analysis suggests).

Despite his deteriorating health, the mathematical output was staggering. Working with Hardy, Ramanujan produced groundbreaking results on the partition function, highly composite numbers, and the distribution of prime numbers. He also continued to work independently, filling notebooks with results on continued fractions, modular equations, infinite series, and q-series. In 1916, he was awarded a Bachelor of Science degree by research for his memoir on highly composite numbers — numbers with more divisors than any smaller positive integer.

In 1918, at the age of 30, Ramanujan was elected a Fellow of the Royal Society — one of the youngest Fellows in history. He was the first Indian to be elected a Fellow of Trinity College. These honors were largely due to Hardy’s tireless advocacy. Hardy understood that formal recognition would help sustain Ramanujan both financially and emotionally, and he fought the institutional barriers with characteristic determination.

Ramanujan returned to India in February 1919, gravely ill, and died on 26 April 1920 at the age of 32. In his short life, he compiled roughly 3,900 identities and equations, many of which were entirely original and ahead of their time. Mathematicians are still working through his notebooks today, more than a century after his death.

21.5 The Hardy–Ramanujan Number 1729

One of the most famous anecdotes in mathematical history involves a hospital visit. When Ramanujan fell ill and was convalescing in Putney, Hardy came to visit. As Hardy later recalled:

“I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. ‘No,’ he replied, ‘it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.’”

Indeed:

This has given rise to the taxicab numbers. The general taxicab number $\text{Ta}(n)$ is defined as the smallest positive integer that can be expressed as a sum of two positive cubes in $n$ different ways. Thus $\text{Ta}(1) = 2 = 1^3 + 1^3$ and $\text{Ta}(2) = 1729$.

There is also a related sequence of cabtaxi numbers, which allow the cubes to be positive or negative. The second cabtaxi number is 91, since$91 = 3^3 + 4^3 = 6^3 + (-5)^3$. The anecdote beautifully illustrates Ramanujan’s intimate familiarity with numbers. He did not need to compute; he simply knew the properties of integers as a musician knows melodies.

21.6 Ramanujan’s Pi Formulas

Among Ramanujan’s most spectacular achievements are his formulas for $\pi$. In 1914 he published a series of 17 such formulas in the paper “Modular equations and approximations to $\pi$”. The most famous is:

The convergence rate of this series is extraordinary: each term adds approximately 8 correct decimal digits of $\pi$. The very first term ($k = 0$) gives:

which yields $\pi \approx 3.14159273\ldots$, correct to 6 decimal places from a single term!

Ramanujan also discovered other remarkable series for pi, including:

In 1987, the Chudnovsky brothers (David and Gregory) discovered a related but even faster-converging series:

This adds about 14 digits per term and has been used to compute trillions of digits of $\pi$. In 2020, Timothy Mullican used the Chudnovsky formula to calculate 50 trillion digits of $\pi$. The formula is a direct descendant of Ramanujan’s original, obtained by finding the optimal parameters in Ramanujan’s class of formulas.

The mathematical machinery behind these formulas involves modular equations, singular moduli, and the theory of complex multiplication of elliptic curves. The number 9801 is $99^2$, and 396 is $4 \cdot 99$; these arise from the singular modulus for the discriminant $-4 \cdot 58$. The appearance of such specific numbers reflects deep arithmetic-geometric connections that Ramanujan perceived intuitively. The Chudnovsky number 640320 is similarly connected to the j-invariant of an elliptic curve with complex multiplication by $\mathbb{Q}(\sqrt{-163})$, exploiting the famous fact that $e^{\pi\sqrt{163}}$ is very nearly an integer.

21.7 The Partition Function and the Hardy–Ramanujan Formula

The partition function $p(n)$ counts the number of ways to write the positive integer $n$ as a sum of positive integers, where the order does not matter.

The generating function for $p(n)$ is given by Euler’s product:

Before Hardy and Ramanujan, there was no formula or even a good approximation for $p(n)$. Their celebrated 1918 paper “Asymptotic Formulae in Combinatory Analysis” was a triumph of the newly invented circle method. The key insight was to express $p(n)$ as a Cauchy integral and then evaluate it by analyzing the behavior of the generating function near roots of unity.

Derivation Sketch: The Circle Method for Partitions

The derivation begins with Cauchy’s integral formula. Setting $q = e^{2\pi i \tau}$ and writing $F(\tau) = \prod_{k=1}^{\infty}(1 - e^{2\pi i k \tau})^{-1}$ for the generating function, we have:

where the contour is a small circle around the origin. Near a rational point $\tau = h/k$ (with$\gcd(h,k) = 1$), the function $F(\tau)$ has a known asymptotic expansion involving the Dedekind eta function. The dominant contribution comes from $\tau$ near 0 (i.e., $h/k = 0/1$), which gives the main exponential term. The contributions from other rational points $h/k$ provide corrections, and summing over all such Farey fractions yields an increasingly precise formula.

Hardy and Ramanujan showed that by taking enough terms, one can approximate $p(n)$ to within$O(n^{-1/4})$, which is less than 1 for sufficiently large $n$ — meaning the formula gives the exact value by rounding. In 1937, Hans Rademacherrefined the method to obtain an exact convergent series:

where $A_k(n)$ is a Kloosterman-type sum involving roots of unity.

Ramanujan’s Partition Congruences

Ramanujan also discovered striking congruence properties of the partition function:

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

For example, $p(4) = 5$, $p(9) = 30$, $p(14) = 135$ are all divisible by 5. Ramanujan proved the first two congruences and conjectured further results. He also gave the beautiful identity:

which makes the divisibility by 5 manifest. These congruences were later explained in terms of the theory of modular forms, connecting combinatorics to deep algebraic and geometric structures. In 2000, Ken Ono proved that such congruences exist for every prime modulus, confirming Ramanujan’s deep intuition.

21.8 Ramanujan’s Continued Fractions

Ramanujan was a master of continued fractions, producing identities of breathtaking beauty and depth. His command of this classical subject surpassed all his predecessors, including Euler and Gauss.

His most celebrated continued fraction result connects an infinite continued fraction to a ratio of infinite products:

This continued fraction has remarkable special values. Ramanujan showed that when $q = e^{-2\pi}$:

involving the golden ratio $\phi = (1+\sqrt{5})/2$. Ramanujan also established that $R(q)$ satisfies a modular equation: if $u = R(q)$ and $v = R(q^2)$, then:

These modular relations allow computation of special values and reveal deep connections to the theory of elliptic functions. Bruce Berndt has systematically proved hundreds of Ramanujan’s continued fraction identities from the notebooks, and many remain active areas of research.

21.9 The Rogers–Ramanujan Identities

The Rogers–Ramanujan identities are among the most beautiful identities in all of mathematics. They were first discovered by L.J. Rogers in 1894, forgotten, then independently rediscovered by Ramanujan before 1913. They were finally proved jointly by Rogers and Ramanujan in 1919.

Combinatorial Interpretation

The first identity has a striking combinatorial meaning. The left side is the generating function for partitions where parts differ by at least 2 (since the $q^{n^2}$ in the numerator enforces gaps). The right side counts partitions into parts congruent to $\pm 1 \pmod{5}$. Thus:

Proof Sketch (via q-difference equations)

One elegant approach defines $G_a(q) = \sum_{n \geq 0} q^{n^2 + an}/((q)_n)$ where$(q)_n = (1-q)(1-q^2)\cdots(1-q^n)$. Then one shows the recurrence:

This, combined with Jacobi’s triple product identity, yields both Rogers–Ramanujan identities. The proof connects the identities to the representation theory of the Virasoro algebra, revealing why the number 5 appears: it relates to the minimal model $\mathcal{M}(2,5)$ in conformal field theory.

The identities have found applications in statistical mechanics(Rodney Baxter’s solution of the hard hexagon model in 1980), representation theory of affine Lie algebras, and conformal field theory. Their depth continues to surprise mathematicians and physicists alike.

21.10 Mock Theta Functions

In his last letter to Hardy (12 January 1920), written just three months before his death, Ramanujan introduced what he called “mock theta functions” — a new class of functions that would puzzle mathematicians for over 80 years.

“I discovered very interesting functions recently which I call ‘Mock’ $\vartheta$-functions. Unlike the ‘False’ $\vartheta$-functions (studied partially by Prof. Rogers in his interesting paper), they enter into mathematics as beautifully as the ordinary theta functions.”

He listed 17 examples organized by “order” (third, fifth, and seventh order). The third-order mock theta functions include:

Ramanujan observed that these functions share some properties with ordinary theta functions (modular forms) but are not themselves modular. They satisfy transformation laws that are “almost” modular, with an error term that Ramanujan could not fully characterize.

For decades, the true nature of mock theta functions remained mysterious. In 2002, Sander Zwegers’ PhD thesis (under Don Zagier at the University of Bonn) finally explained them: they are the holomorphic parts of harmonic Maass forms. Specifically, each mock theta function can be “completed” by adding a non-holomorphic integral involving a unary theta function, producing a function that transforms correctly under the modular group.

This discovery unified Ramanujan’s mock theta functions with the classical theory of modular forms and spawned an explosion of research. Mock modular forms now appear in the theory of black holes (via the work of Dabholkar, Murthy, and Zagier on quantum entropy), moonshine phenomena (Eguchi, Ooguri, and Tachikawa’s discovery of Mathieu moonshine), and combinatorics (overpartitions, unimodal sequences). Ramanujan’s last creation, conceived on his deathbed, turned out to be one of his most far-reaching contributions.

21.11 Ramanujan’s Notebooks: Remarkable Formulas

Ramanujan left behind three notebooks (written before his departure to England) and the “Lost Notebook” (written in his final year). Together they contain approximately 3,900 results. George Andrews and Bruce Berndt devoted decades to systematically proving and contextualizing these results, publishing five volumes on the notebooks and several more on the Lost Notebook. Here are some of the most remarkable formulas.

The work of proving and understanding Ramanujan’s notebook entries continues to this day. As Andrews has said, the notebooks are “a garden of mathematical delights that will continue to blossom for centuries.”

21.12 Modular Forms and the Ramanujan Tau Function

Modular forms are among the most important objects in modern number theory. They are functions on the upper half-plane $\mathcal{H} = \{\tau \in \mathbb{C} : \text{Im}(\tau) > 0\}$ that satisfy a remarkable transformation law under the action of the modular group.

Ramanujan’s most famous contribution to this theory is the Ramanujan tau function $\tau(n)$, defined by the expansion of the discriminant modular form:

where $q = e^{2\pi i \tau}$. The first few values are $\tau(1) = 1$, $\tau(2) = -24$, $\tau(3) = 252$, $\tau(4) = -1472$, $\tau(5) = 4830$.

Ramanujan made three deep conjectures about $\tau(n)$:

1. Multiplicativity: $\tau(mn) = \tau(m)\tau(n)$ when $\gcd(m,n) = 1$. Proved by Mordell (1917).
2. Recurrence for prime powers: $\tau(p^{a+1}) = \tau(p)\tau(p^a) - p^{11}\tau(p^{a-1})$. Also proved by Mordell.
3. The Ramanujan conjecture: $|\tau(p)| \leq 2p^{11/2}$ for all primes $p$. This was vastly more difficult and was finally proved by Pierre Deligne in 1974 as a consequence of his proof of the Weil conjectures, work that earned him the Fields Medal (1978).

The Ramanujan conjecture (now Ramanujan–Petersson conjecture) has profound generalizations in the Langlands program, one of the deepest ongoing research programs in mathematics. Ramanujan’s intuition, formulated in 1916 with minimal formal education, pointed directly at the heart of 21st-century number theory.

21.13 The Lost Notebook

In 1976, the mathematician George Andrews of Penn State was visiting the Wren Library at Trinity College, Cambridge, when he discovered a collection of about 130 pages of Ramanujan’s unpublished work buried in a box of papers belonging to the late G.N. Watson. These pages, written during the last year of Ramanujan’s life (1919–1920), contained over 600 formulas, most without proof.

Andrews immediately recognized the treasure: the “Lost Notebook” contained material at least as remarkable as the published notebooks. The central topic was mock theta functions, but the pages also contained results on q-series, Rogers–Ramanujan type identities, partial theta functions, and definite integrals. The Lost Notebook has been described as the mathematical equivalent of discovering Beethoven’s tenth symphony.

Andrews and Bruce Berndt published a series of volumes providing proofs and commentary for every entry in the Lost Notebook. Their work, spanning decades, has connected Ramanujan’s results to modern developments in combinatorics, representation theory, statistical mechanics, and algebraic geometry. Many of the results in the Lost Notebook anticipated developments that would not occur in mainstream mathematics until 60 or 80 years later.

21.14 Legacy — Intuition, Rigor, and the Mathematical Enterprise

The stories of Ramanujan and Hardy illuminate different facets of the mathematical enterprise. Ramanujan represents the power of raw mathematical intuition — an almost superhuman ability to perceive patterns and relationships in numbers. Hardy represents the tradition of rigorous proof and the insistence that mathematics must meet the highest standards of logical certainty.

Ramanujan’s story is also a sobering reminder of how much genius may be lost when talent lacks opportunity. Had he been born into wealth, or in a country with a developed mathematical tradition, his output might have been even more extraordinary. Conversely, his very isolation may have contributed to his originality: freed from conventional training, he developed his own methods and attacked problems from unexpected angles. As Hardy wrote: “We owe it to him that he was not entirely self-taught. But it would have been much better for mathematics if he had been discovered and properly trained at a much earlier stage.”

The Hardy–Ramanujan partnership showed that the most productive mathematical work often emerges from the collaboration of complementary minds. Hardy’s rigor gave form and proof to Ramanujan’s inspired conjectures; Ramanujan’s vision opened domains that Hardy’s methods could then explore. Their work on the partition function, in particular, created the circle method that would become one of the most powerful tools in analytic number theory, with applications extending far beyond the original problem.

Hardy himself remained haunted by Ramanujan’s early death. In his later years, he rated his own contributions to mathematics as modest compared to what Ramanujan might have achieved. He devoted considerable effort to publishing and promoting Ramanujan’s work, ensuring that it would not be forgotten. Hardy’s Twelve Lectures on Subjects Suggested by His Life and Work (1940) remains an essential introduction to Ramanujan’s mathematics.

Today, Ramanujan’s influence pervades many areas of mathematics: analytic number theory, combinatorics, the theory of modular forms and q-series, representation theory, mathematical physics, and even string theory. The Ramanujan Journal (founded 1997) publishes research inspired by his work. December 22, his birthday, is celebrated as National Mathematics Day in India. His notebooks continue to yield new results and inspire new mathematics. As Freeman Dyson wrote: “Whenever I am amazed by a result of modern mathematics, I find that Ramanujan knew it already.”