Srinivasa Ramanujan

Srinivasa Ramanujan
	Srinivasa Ramanujan, Indian mathematician
Tradition	Mathematics, Number theory, Analysis
Influenced by	G. H. Hardy, Leonhard Euler (indirectly through study of his work)
Lifespan	1887–1920
Notable ideas	Ramanujan prime; Ramanujan theta function; highly composite numbers; infinite series and continued fractions
Occupation	Mathematician
Influenced	G. H. Hardy, Modern number theory, Mathematical analysis
Wikidata	Q83163

Srinivasa Ramanujan (1887–1920) was a self-taught Indian mathematical genius known for his profound contributions to number theory, analysis, and infinite series. He produced nearly 3,900 mathematical results (mostly formulas and identities) in his brief life, many of which were entirely novel and opened up whole new areas of research. Ramanujan’s intuitive approach often led to breakthroughs that Europeans considered unsolvable with the methods of the day. His work ranged from surprising congruences in partition functions to rapidly converging series for π and new kinds of special functions. After the publication of his first papers in India in 1911, he corresponded with G. H. Hardy at Cambridge, who arranged for Ramanujan to live and work in England from 1914 to 1919. Ramanujan’s legacy endures in ongoing research, in mathematical concepts named after him, and in cultural tributes ranging from a journal and prizes bearing his name to films and national holidays celebrating his life.

Early life and education

Ramanujan was born on 22 December 1887 in Erode (then Mysore Presidency), in a Tamil Brahmin family. His father Kuppuswamy Srinivasa Aiyangar was a clerk in a sari shop, and his mother Komalatammal was a housewife. The family soon moved to the temple town of Kumbakonam in present-day Tamil Nadu, where Ramanujan’s talent for mathematics became evident at an early age. By age 10 he had topped district examinations and earned a gold medal for mathematics. He taught himself advanced mathematics books beyond his school curriculum, eventually digesting G. S. Carr’s *Synopsis of Elementary Results in Pure and Applied Mathematics* (containing 5,000 theorems). Carr’s compendium (covering material up to about 1860) stimulated Ramanujan to use each result as a springboard to discover many of his own theorems.

In 1904, Ramanujan received a scholarship to study at Government Arts College in Kumbakonam. However, he was so single-mindedly devoted to mathematics that he neglected most other subjects. He failed most of his non-mathematical courses and thus lost the scholarship within a year. He briefly attended Pachaiyappa’s College in Madras (Chennai) around 1906, again failing several subjects despite excelling in mathematics. After leaving college without a degree, he continued pure mathematical research alone while living in poverty. He took a clerical job at the Madras Port Trust in 1912 (at the entrance exam level) to support himself, but most of his spare time was devoted to exploring number theory and series.

Despite his difficult circumstances, Ramanujan published his first paper in 1911 on Bernoulli numbers in the Journal of the Indian Mathematical Society. Over the next few years he produced short papers on topics like algebraic equations, divisors of integers, and question-and-answer problems posed by Indian mathematics journals. By 1912 his remarkable results began to attract attention. He married Janakiammal in 1909 (a marriage arranged by his family when she was a child) and continued to seek a way to pursue mathematics full-time. His poverty and lack of formal credentials made it hard to find academic support, but in 1913 he approached the distinguished Indian mathematician R. Ramachandra Rao, secretary of the Indian Mathematical Society. Rao was impressed by Ramanujan’s abilities and provided some financial assistance and encouragement. At Ramanujan’s request, Rao helped arrange correspondence with leading mathematicians in England.

Early in 1913 Ramanujan sent samples of his mathematical work (nine typed pages of formulas on divergent series and elliptic integrals) to several British mathematicians. University College London professor M. J. M. Hill commented that the papers were “riddled with holes” in reasoning, though he said Ramanujan had some ability. Many mathematicians ignored the letters. However, G. H. Hardy at Cambridge recognized the extraordinary creativity in them. At first Hardy thought the work might be a joke or hoax, but he identified some results and eventually concluded that Ramanujan’s theorems, however unfamiliar in style, could hardly have been made up. In January 1914 Hardy invited Ramanujan to Cambridge to study and collaborate with him, arranging travel support through Trinity College.

Ramanujan left India for England in March 1914. The journey was complicated by World War I and Ramanujan’s strict vegetarian diet (as a devout Brahmin), but he arrived in Cambridge in April. Hardy and a colleague, E. H. Neville, helped him settle into Trinity College. Without a formal degree, Ramanujan was allowed to enroll by research. He suffered from homesickness and later illness in the cold climate, but he continued his work. In March 1916 he was awarded a Cambridge BA by research for his dissertation on highly composite numbers (integers with more divisors than any smaller number). By that time, he had produced many papers and the famous notebooks (handwritten collections of theorems) that he would continue filling at Cambridge.

Major works and ideas

Ramanujan’s published and unpublished results cover a wide range of mathematics, often going far beyond the frontier of his time. His creativity can be grouped into several major themes:

Partition functions and congruences. One of Ramanujan’s most famous subjects was the partition function $p(n)$, which counts the ways of writing $n$ as a sum of positive integers (order unimportant). In 1917–1918, Hardy and Ramanujan found an asymptotic formula for $p(n)$ (later refined by Rademacher) showing that $p(n)$ grows roughly like $$p(n)\sim \frac{1}{4\sqrt{3}\,n} \exp\!\big(\pi\sqrt{\tfrac{2n}{3}}\big)\quad\text{as }n\to\infty.$$ This formula was groundbreaking in combinatorics. Ramanujan also discovered remarkable arithmetic congruences for $p(n)$: for example, $p(5n+4)$ is divisible by 5, $p(7n+5)$ by 7, and $p(11n+6)$ by 11 for all integers $n\ge0$. (These congruences were proven with modular forms later.) He went on to generalize these ideas and even defined the notion of a Ramanujan prime, related to the distribution of prime numbers.

Modular forms and theta functions. Ramanujan had deep insights into modular functions and elliptic integrals. He discovered highly nontrivial formulas for the Fourier coefficients of modular forms, including what are now called Ramanujan’s tau-function and the partition theta function. In one of his final works (1919–1920), he introduced a mysterious class of $q$-series he called “mock theta functions.” These functions generalize classic theta functions but do not fit into the usual theory of modular forms. Ramanujan listed 17 examples of mock theta functions in his last letter. These puzzled mathematicians for decades, until the 2000s when a new theory showed they are related to harmonic weak Maass forms. Modern research (led by K. Ono and others) has confirmed that Ramanujan’s mock-theta ideas have far-reaching connections, even to areas like string theory and black hole physics, demonstrating the foresight of his work.

Infinite series and products for $\pi$ and other constants. Ramanujan discovered many surprisingly rapid infinite series for $\pi$ and related constants. In a 1914 paper, he published 17 new series for $1/\pi$ (based on modular equations), many of which converge extraordinarily quickly. For example, one of his series implies the approximation $$\pi\approx\frac{9801\sqrt{2}}{4412},$$ which is correct to six decimal places using only the first term. His series became the foundation of modern algorithms for computing $\pi$ to billions of digits (the Chudnovsky brothers’ formulas are direct descendants of Ramanujan’s work). Ramanujan also studied divergent series in a novel way. He invented a technique now called Ramanujan summation that assigns finite values to certain divergent series (like $1 + 2 + 3 + \cdots = -\tfrac{1}{12}$ in his regularized sense). Although controversial at the time, Ramanujan’s summation methods later found use in physics (especially in quantum field theory) where similar divergent sums arise.

Diophantine identities and constants. Ramanujan often produced remarkable identities involving integers and constants. For instance, he discovered near-integer values of $e^{\pi\sqrt{n}}$ for certain $n$; famously, $e^{\pi\sqrt{163}}$ differs from an integer by less than 0.00000000014. He also identified special constants like the Ramanujan–Soldner constant and provided formulas involving the Euler ϕ and σ functions (sums of divisors). His early notebooks include many striking algebraic identities. Together with mathematicians like G. H. Hardy, he gave precise formulas for arithmetic functions (for example, expressing divisor sums via mohgorapolic forms). Many of these results were initially conjectural and only proved rigorously decades later. His ability to spot patterns in numbers was so sharp that G. Hardysy once remarked that Ramanujan’s conjectures “defeated me completely; I had never seen anything in the least like them before,” concluding that no one could invent such results unless they were true.

Continued fractions and special integrals. Ramanujan showed extraordinary expertise with continued fractions, a branch of analysis. He found general continued fraction expansions for functions and constants that were unknown before. He also evaluated many definite integrals (especially elliptic and modular integrals) in closed form. For example, he solved integrals involving the gamma function and elliptic integrals that had challenged mathematicians. These results often appeared without derivation in his papers.

Together, these ideas from Ramanujan opened new lines of inquiry. The Rogers–Ramanujan identities (two famous $q$-series identities relating partitions to modular functions) originated from Ramanujan’s work and are now fundamental in combinatorics. In number theory, his prime-counting conjectures and tau-function congruences inspired later giants like Deligne and Serre. His bold Congruences for partitions anticipated the modern theory of modular forms. In summary, Ramanujan’s major contributions span pure number theory, analysis, and even mathematical physics, and many carry his name (e.g. Ramanujan’s graphs, Ramanujan’s constant, Ramanujan prime, Ramanujan summation, etc.).

Method and philosophy

Ramanujan’s approach to mathematics was exceptional and often mystical. Unlike a typical mathematician, he did not focus on formal proofs at first; he recorded results that came to him with uncanny intuition. He often attributed his inspiration to the Hindu goddess Namagiri of Namakkal, saying that she appeared in his dreams and wrote mathematical formulas on a divine scroll for him. As he famously said, *“An equation for me has no meaning unless it expresses a thought of God.”* This deep spirituality infused his work: he believed that mathematical insight was revealed to him, not invented from step-by-step derivations.

In practical terms, Ramanujan rarely provided detailed proofs of his theorems. He claimed, for example, that he could see that an identity was true without working out every step. This tendency initially puzzled Western mathematicians. G. H. Hardy later commented that many of Ramanujan’s formulas were so ingenious that, “if they were not true, no one would have the imagination to invent them.” In other words, Ramanujan’s creativity was so rich that mathematicians had to accept his results as correct and then work backward to prove them rigorously. This style sometimes drew criticism; some reviewers found his papers “riddled with holes” because they lacked standard justifications. However, most of his results were later confirmed, showing that his intuition was overwhelmingly reliable.

Ramanujan was largely self-educated and worked in solitude before Cambridge. He often did all computation and deduction in his head or on paper sheets, filling pages of notebooks with formulae. He showed little interest in established literature; when he arrived in England, he discovered much of modern analysis on the spot. Hardy once tried to teach him complex analysis and gamma functions, but Ramanujan continued generating new theorems at such a fast pace that he barely paused to learn existing theory. According to biographers, his collaborator John Littlewood found that every time he explained a new concept, Ramanujan would invent fresh results immediately, so that teaching became impossible.

Despite his unconventional methods, Ramanujan’s work was not careless. He investigated every result he wrote down, often checking terms to dozens of decimal places. His notebooks show a systematic inventory of results – some with sketches of proof – even if details were omitted. His intuition-driven process resembles pattern-finding or conjecturing new identities based on experimental observation, then later validating them. In modern terms, one might say he had a remarkable ability to navigate the “mathematical landscape” by intuition and then, sometimes much later, provide a formal explanation. This blend of divine inspiration and analytical skill is why mathematicians often describe Ramanujan as having a form of mathematical genius beyond conventional training.

Influence and reception

In his lifetime and especially after his death, Ramanujan was recognized as one of the greatest mathematical talents. Early on, Cambridge mathematicians Hardy and Littlewood were astonished by Ramanujan’s originality. Hardy famously wrote that there were three types of mathematicians: those who see things to be true, those who prove things, and a third category which includes Ramanujan. By the time he was 30, Ramanujan had been elected a Fellow of the Royal Society of London (in 1918) and a Fellow of Trinity College, Cambridge – honors remarked upon as extraordinary for an Indian mathematician of that era. He was the first native Indian to become a Fellow at Trinity, and only the second Indian ever elected to the Royal Society. These accolades reflected the esteem in which his peers held him.

World-wide, mathematicians compared Ramanujan to legends like Euler and Gauss. An encyclopedic account in 1935 by Hardy and others stressed that Ramanujan’s results “must be true” and that his contributions “defeated” conventional mathematics. After his death, much attention turned to his notebooks. Three volumes of Ramanujan’s published papers appeared (1927–1930), and the rest of his notebooks were edited and released over the decades (with Bruce Berndt and others publishing extensive commentaries). Perhaps most famously, his “Lost Notebook” – a sheaf of papers from his last year – was rediscovered in 1976 by mathematician George Andrews. The Lost Notebook, containing hundreds of unproved formulas, has since fueled ongoing research: dozens of papers and five volumes of annotated results by Andrews and Berndt have been produced. These efforts gradually verified or placed Ramanujan’s final findings in context, demonstrating that even his unexplained snippets often held deep truths.

Ramanujan’s legend and story have also prominently entered popular culture. In India, his birthday (22 December) was declared National Mathematics Day in 2012. Statues of Ramanujan stand at mathematics institutes in India, and multiple Indian postage stamps have been issued in his honor (in 1962, 2012, 2013 and 2017, marking anniversaries of his birth and death). The Ramanujan Journal, founded in 1994, publishes research inspired by his work. Internationally, books have chronicled his life, most notably Robert Kanigel’s The Man Who Knew Infinity (1991), which inspired the major motion picture of the same name in 2015 (starring Dev Patel as Ramanujan). Many competitions and conferences in mathematics also invoke his name. Tributes note that even today, decades after his passing, researchers are finding value in his ideas – for example, in 2012 a team led by Ono used Ramanujan’s mock theta functions to make advances related to black hole entropy. In short, Ramanujan’s contributions have had a lasting impact on mathematics and scientific culture.

Critiques and debates

Ramanujan’s methods and story have also been the subject of debate. Some critics have cautioned against over-romanticizing his mysticism at the expense of mathematical substance. Because Ramanujan credited dreams for inspiration, some have suggested that emphasizing this angle distracts from the intellectual work of proving his results. Others argue that his reliance on intuition underscores a valid creative process: he saw mathematics “in the mind’s eye” before formalizing it. This discussion reflects a broader debate about the role of intuition versus rigor in mathematics.

In academic circles, early reception of Ramanujan’s work was mixed. As noted, some mathematicians initially dismissed his letters or found them lacking in standard proofs. Hardy and others had to carefully verify many of his claims. In fact, a lecturer at University College London reportedly read Ramanujan’s papers and termed them full of conjectures, doubting their value. Nevertheless, once key results (such as his partition congruences and series for π) were validated, most skepticism faded. By contrast, everyone acknowledged Hardy’s own admiration: he wrote Ramanujan’s obituary and biography, stating that the raw originality of his material overcame all doubts.

Some modern commentators have pointed out that Ramanujan’s mathematical outlook was shaped by the knowledge available in India in his youth. He was largely unaware of developments after 1900 (like rigorous analysis and topology) until he went to England. For instance, he knew nothing of Cauchy’s Theorem or the latest theories of quadratic forms when he left for Cambridge. This has led to reflection on how unorthodox his background was – he essentially rediscovered or independently developed concepts in isolation. Some scholars have even debated how to categorize his work: Was he discovering hidden genius mathematics, or utilizing deep but classical formulas rambling as if new? The consensus view now is that he contributed original ideas; many results bear his name.

Moreover, there is occasional debate about the particulars of his biography. For example, some historical analyses scrutinize his cause of death (once thought to be tuberculosis, now believed to be hepatic amoebiasis from earlier dysentery). Others discuss how his living conditions and diet affected his health and work in Cambridge. These questions, though biographical, sometimes influence interpretations of his productivity and struggles during those years.

Overall, however, no serious controversy tarnishes Ramanujan’s mathematical legacy. The main debates are over interpretation of his personal narrative (religious inspiration, the role of intuition) rather than the validity of his mathematics, which has been overwhelmingly confirmed.

Legacy

Ramanujan’s legacy is profound and multifaceted. Mathematicians today still actively study his formulas and name many objects after him. For instance, the Ramanujan tau function (arising from the expansion of the discriminant modular form) was proved years later to satisfy deep conjectures (Deligne’s proof of the Weil conjectures in 1973 showed Ramanujan’s conjectured properties of its values actually held). The Ramanujan primes and Ramanujan–Nagell equation remain the subjects of research, and the Landau–Ramanujan constant (describing the density of numbers that are a sum of two squares) originated from speculation partly due to Ramanujan’s influence. In applied mathematics and physics, Ramanujan’s ideas appear in surprising ways: the assignment of $\sum n = -1/12$ is used in string theory calculations, and mathematicians and physicists still explore his mock theta functions for links to black hole entropy and moonshine theory.

In the education of mathematicians, Ramanujan is held up as a symbol of pure intuition and creative genius. His story (a poor clerk who communicated with Western mathematics leaders and produced brilliant results in a few years) is celebrated especially in India and among students from underrepresented backgrounds, as an example of exceptional talent overcoming obstacles. India honors his memory with events and institutions: the Ramanujan Math House in Kerala and the Ramanujan Institute for Advanced Study in Mathematics (in Chennai) are named for him. The SASTRA Ramanujan Prize (awarded annually to an outstanding mathematician under age 32) commemorates the age at which Ramanujan died. In addition, his face has appeared on Indian currency and postage stamps, and Google marked his 125th birthday with a doodle.

Finally, Ramanujan’s own writings continue to be mined for discoveries. The Ramanujan Journal founded in 1994 publishes research connected to his work. Scholars like Bruce C. Berndt have spent decades proving or extending entries from Ramanujan’s notebooks, publishing comprehensive volumes on Ramanujan’s papers. As recently as the 2010s, researchers reported that even remarks in his annulments of ordinary functions pointed to nontrivial theorems that had been missed. In this way, the Ramanujan notebooks (including the Lost Notebook) have given rise to new mathematics long after his death, underscoring that his creative vision had depths still being fathomed.

In popular culture, his life continues to resonate: plays, books, and movies portray his friendship with Hardy, and essays reflect on his blend of spiritual insight and mathematical brilliance. Even in the 21st century, Srinivasa Ramanujan is more than a historical figure — his methods inspire ongoing research, his story inspires mathematicians, and his discoveries live on in the structure of mathematics.

Selected works

Some properties of Bernoulli's numbers (1911) – Ramanujan’s first published paper, establishing early results on Bernoulli numbers and asymptotics of partition functions.

Highly Composite Numbers (1915, Proc. London Math. Soc.) – A long paper giving detailed properties of integers with more divisors than any smaller number; this formed Ramanujan’s Cambridge dissertation and coined the idea of “highly composite” numbers.

Modular equations and approximations to π (1914, Quarterly J. Math.) – A celebrated paper containing 17 new rapidly convergent series for $1/π$, which later enabled high-precision computations of π.

On certain arithmetical functions (Hardy & Ramanujan, 1918, Phil. Trans. Royal Soc.) – A joint paper proving asymptotic formulas for the partition function and divisor functions, seminal in analytic number theory.

Notebooks of S. Ramanujan (edited by G. N. Watson and B. M. Wilson, 1923–1927) – Published collections of Ramanujan’s handwritten results; these books compiled much of his published and unpublished work.

Ramanujan’s Lost Notebook (edited by G. E. Andrews and B. C. Berndt, Parts I–V, 2005–2018) – A five-volume series of annotated proofs for the hundreds of formulas Ramanujan left without proof at the end of his life (written down c.1919–1920).

The Man Who Knew Infinity (1991) – A popular biography by Robert Kanigel (not by Ramanujan, but illustrative of works inspired by him).

(These examples highlight key publications and posthumous collections of Ramanujan’s work.)

Timeline

1887 – Born 22 December in Erode, Madras Presidency (now Tamil Nadu), India.

1893 – Family moves to Kumbakonam; Ramanujan begins school education.

1897 – At age 9, passes district’s primary exam with top marks.

1903 – Obtains Synopsis of Pure and Applied Mathematics by G. S. Carr and studies it intensively.

1904–1905 – Wins a department scholarship to Government Arts College, Kumbakonam, but loses it by focusing only on mathematics.

1909 – Marries Janakiammal on 14 July. Began seeking recognition from the wider mathematical community.

1911 – Publishes first paper (Bernoulli numbers) in the Journal of the Indian Mathematical Society. Accepts a clerk’s job at Madras Port Trust.

1913 – Begins correspondence with G. H. Hardy (letter dated January); Ramanujan’s work is studied in Madras and introduced to British mathematicians.

1914 – Arrives in Cambridge, England (April). Meets Hardy and E. H. Neville. Begins five-year collaboration at Trinity College, Cambridge.

1916 – Awarded a BA (by research) from Cambridge for his work on highly composite numbers (16 March); elected Fellow of the Cambridge Philosophical Society.

1917 – Collaborates with Hardy on partitions and number theory; Hardy and Ramanujan publish foundational results on the partition function.

1918 – Elected Fellow of the Royal Society (February); chosen Fellow of Trinity College, Cambridge (February).

1919 – Due to failing health (tuberculosis and other ailments), returns to India in January. Continues to work and write until his final illness.

1920 – Ramanujan dies on 26 April in Kumbakonam at age 32; a handwritten “lost notebook” containing work from his last year remains to be catalogued by his brother.

1927 – Cambridge University Press publishes Collected Papers of S. Ramanujan, preserving his legacy.

1976 – Mathematician George E. Andrews rediscovers Ramanujan’s Lost Notebook in the Trinity College archives, generating renewed interest and research.

1991 – Biographer Robert Kanigel publishes The Man Who Knew Infinity, drawing further international attention to Ramanujan’s life.

1994 – The Ramanujan Journal is founded to publish work in areas influenced by Ramanujan.

2012 – Indian government declares 22 December (Ramanujan’s birthday) as National Mathematics Day; stamp issued.

2015 – The Man Who Knew Infinity film is released (Dev Patel as Ramanujan).

2018 – The University of Cambridge and SASTRA University (India) celebrate Ramanujan’s 125th birth anniversary (a statue is unveiled in New Delhi on 28 Feb).

Conclusion

Srinivasa Ramanujan rose from humble beginnings to leave an indelible mark on mathematics. His prodigious output – much of it produced in his twenties – continues to inspire and astonish. Many of his findings were so far ahead of their time that they were only fully understood and proved long after his death. To this day, mathematicians open his notebooks to find new ideas, and his name is synonymous with deep intuition and creativity. Ramanujan’s fusion of cultural belief and scientific genius makes him a unique figure: a mathematician who saw beauty in numbers as if guided by a higher power. Whether through the elegant formulas he discovered or the story of his extraordinary life, Ramanujan’s legacy lives on as one of the brightest chapters in the history of mathematics.