Mock Theta Functions: Ideas Ahead of His Time
Mathematical concepts that took 80 years to fully understand
Discover Ramanujan's mock theta functions, mysterious mathematical objects whose true nature was not understood until the twenty-first century.
Mock Theta Functions: Ideas Ahead of His Time
On January 12, 1920, a twenty-eight-year-old, very sick man in a small house in Kumbakonam dictated one last letter to his collaborator G. H. Hardy in Cambridge. He had come home from England ten months earlier. Tuberculosis, or possibly hepatic amoebiasis, had reduced him to skin over bone. He would die in three months. The letter, barely ten handwritten pages, contained no complaints, no farewells, and only the briefest gossip. What it contained instead was a new class of mathematical object that he had invented and named himself: mock theta functions. He listed seventeen of them, arranged by 'order' into groups of three, five, and seven, and offered almost no explanation of what the words 'mock theta' were supposed to mean.
For eighty-two years, that letter was one of the great unsolved riddles of modern mathematics.
A Letter Without a Theory
Ramanujan was accustomed to writing down results that other mathematicians did not yet know how to reach. He had done it in his Indian notebooks, in his first letter to Hardy in 1913, and throughout his Cambridge years. By the standards of his own earlier output, the January 1920 letter was unusual for a different reason. It did not just state results. It introduced a name for a kind of function, gave examples, hinted at properties, and then stopped. The letter said, in effect, that these things exist, they are important, they are not theta functions even though they look like theta functions from some angles, and here are seventeen of them. Go figure out why.
Hardy filed the letter, spoke of it after Ramanujan's death as part of the 'final' work, and helped ensure its contents were published. For most of the twentieth century, the mock theta functions sat in the literature as an exotic collection of formulas that clearly pointed at something and just as clearly were not yet understood. Number theorists played with them. Physicists noticed them. Nobody could say what they really were.
Theta Functions: The Thing They Mock
Theta functions are some of the most studied objects in classical analysis. Jacobi theta functions, Eisenstein series, and their cousins possess a property called modularity. If you plug a complex variable through a particular family of substitutions (the modular group), the function transforms in a predictable, elegant way. Modularity is to theta functions what symmetry is to crystals: the hidden skeleton that makes a vast universe of identities fall out automatically.
A mock theta function looks as if it should also be modular. Its power-series coefficients grow at the right rate; its asymptotic behaviour near the cusps of the modular domain mimics a real theta function. But it is not modular. Nobody in 1920, and nobody for decades afterwards, could give a rigorous theory of what failure of modularity was happening, or how consistent that failure was across Ramanujan's seventeen examples.
2002: Sander Zwegers and the Maass Form Bridge
The breakthrough came from an unexpected place. In 2002, a Dutch graduate student named Sander Zwegers submitted a PhD thesis at Utrecht University titled simply 'Mock Theta Functions'. Zwegers showed that each of Ramanujan's mock theta functions was the holomorphic part of a new kind of object, a harmonic Maass form. A harmonic Maass form is modular, in a generalized sense, once you take it together with a non-holomorphic completion. The mock theta function itself is what you get when you strip the completion away: it is the holomorphic shadow of a more complete, modular object that Ramanujan never wrote down.
Ramanujan could not have invented harmonic Maass forms in 1920. The theory of Maass forms was developed by Hans Maass in the 1940s, and the harmonic version that Zwegers needed was not pinned down until the 1990s. What Ramanujan had done, alone in Kumbakonam, was write down seventeen examples of the holomorphic shadows of something that mathematics did not yet have the vocabulary to name.
Black Holes, Partitions, Physics
Once Zwegers had handed the community a theory, the applications came quickly. Ken Ono, Kathrin Bringmann, and Jan Bruinier used mock modular forms to finally prove Freeman Dyson's 1944 conjectures about the rank and crank of integer partitions, closing a sixty-year-old open problem with Ramanujan's own tools. String theorists, led by Atish Dabholkar, Sameer Murthy, and Don Zagier, showed that mock modular forms compute the exact microstate degeneracies of certain supersymmetric black holes: the same functions Ramanujan listed on his deathbed turned out to be the natural mathematical language for counting quantum states of black holes. In 2013, Ono and colleagues confirmed that every known mock theta function fits Zwegers' framework, closing the 1920 letter as a complete research object.
Seeing Across the Gap
Ramanujan's mock theta functions raise a question that does not go away however long you stare at it. How did a twenty-eight-year-old autodidact in South India, working without modular forms theory, without harmonic analysis, without string theory, without black hole thermodynamics, write down seventeen perfectly chosen examples of the holomorphic parts of harmonic Maass forms? He could not have known what he was doing in our sense. He seems to have known what he was doing in his own sense: an intuitive grasp of which kinds of formal expressions carried real content and which did not.
The lesson of his last letter is not that intuition can replace rigor. It is that intuition sometimes travels faster than rigor, and that the job of later mathematicians is to catch up. Mock theta functions took eighty-two years to catch up. They are catching up still.
In the small house in Kumbakonam, the dictation ended. Ramanujan sealed the letter, addressed it to Cambridge, and would die in fourteen weeks. Eighty-two years passed before the rest of the world finished reading what he had written that morning.
Key figures
Srinivasa Ramanujan
December 22, 1887 to April 26, 1920, late British India
Sander Zwegers
Born 1975, active present-day
G. H. Hardy
1877 to 1947, late Victorian and early twentieth-century British mathematics
Case studies
The Last Letter: January 12, 1920
On January 12, 1920, Ramanujan, bedridden in Kumbakonam and dying of what his doctors called tuberculosis, dictated or wrote a letter to G. H. Hardy at Trinity College, Cambridge. The letter was short and almost entirely mathematical. In it he defined what he called 'mock theta functions', listed seventeen examples organized by 'order' (three of order three, ten of order five, and four of order seven), and stated several identities relating them without supplying proofs. He did not explain why he had chosen the word 'mock', did not offer a general theory, and did not indicate how he had arrived at the particular examples. He died one hundred and four days later, on April 26, 1920, at the age of thirty-two.
Ramanujan consistently described his mathematics as something received rather than something invented, attributing formulas to the goddess Namagiri Thayar of Namakkal. The Upanishadic tradition makes room for exactly this mode of knowing: rahasya, the secret teaching that is glimpsed before it can be fully formulated. The January 1920 letter reads as a rahasya handed forward. The sender is too weak to complete the teaching; the receiver is expected to preserve it until a later generation can unfold it.
Hardy preserved the letter and had its mathematical content published posthumously in Ramanujan's Collected Papers in 1927. The seventeen functions were studied by George Watson, Lucy Slater, George Andrews, and Bruce Berndt across the twentieth century, producing many specific identities but no overarching theory. The 1976 discovery of Ramanujan's 'Lost Notebook' at the Wren Library in Cambridge added more mock theta functions and more unanswered questions. The letter remained a puzzle for eighty-two years.
A dying mathematician chose, in his final months, to invest his remaining strength in an idea he knew he could not complete. The choice shows what he thought mathematics was for: not a personal portfolio but a gift to future problem-solvers.
The letter was written one hundred and four days before Ramanujan's death at age thirty-two. It contained approximately seventeen mock theta functions and the first use of the phrase in any mathematical source.
Sander Zwegers, 2002: A PhD Thesis Closes an 82-Year Gap
In 2002, Sander Zwegers, a PhD student at Utrecht University in the Netherlands working under Don Zagier, defended a thesis titled 'Mock Theta Functions'. The thesis proved that each of Ramanujan's seventeen mock theta functions could be 'completed' to a harmonic Maass form, a new kind of non-holomorphic modular object whose theory had been developed only in the 1990s. In other words, each mock theta function turned out to be the holomorphic shadow of a complete, hidden modular structure. The strange name 'mock' that Ramanujan had chosen, and had never explained, suddenly made technical sense: the functions mimic modularity without being modular, because they are the visible part of something that is modular.
Zwegers' thesis is a textbook example of pratibha being caught up with by yukti. Ramanujan's flash of insight (the darshana he attributed to Namagiri) had picked the right seventeen examples before there was a language for why they were right. Zwegers, working with tools Ramanujan never had, supplied the language. The Kerala school model of knowledge, in which intuition and demonstration are partners separated sometimes by generations, describes this episode more naturally than the usual Western 'lone genius' framing.
Zwegers' thesis effectively launched the modern theory of mock modular forms. Within a decade, Ken Ono, Kathrin Bringmann, and Jan Bruinier had used the framework to prove Freeman Dyson's 1944 partition conjectures, a sixty-year-old open problem. String theorists extended the framework to black hole physics. By 2013, the community regarded the theory of mock modular forms as one of the most active areas of twenty-first-century number theory. A piece of Ramanujan's work that had sat dormant for eighty-two years became, within a decade, a central field.
The usable life of a great idea can easily exceed a human lifespan. Preserve unexplained insights carefully; you may not be the one who gets to cash them in.
From Namakkal to Black Holes: Counting Quantum States
In a series of papers beginning around 2010, Atish Dabholkar, Sameer Murthy, and Don Zagier showed that mock modular forms, the framework Zwegers had extracted from Ramanujan's mock theta functions, compute the exact degeneracies of states of certain supersymmetric black holes in string theory. Earlier work by Ashoke Sen and others had already shown that the entropy of such black holes can be derived from counting microstates; the Dabholkar-Murthy-Zagier contribution was to show that the fine-grained corrections to these counts are packaged precisely by mock modular forms. The very functions Ramanujan had listed in his January 1920 letter reappeared in the mathematical apparatus of quantum gravity.
Indian metaphysics has a long habit of treating the counting of states as a spiritually serious activity, from the Samkhya enumeration of the tattvas to the Jain theory of the infinite hierarchies of beings. Ramanujan's work on partitions, and through partitions on mock theta functions, sits inside the same lineage. It is fitting that the functions a devotee of Namagiri Thayar listed in his dying weeks turned out, a century later, to count the quantum states of the most extreme gravitational objects in the universe.
Mock modular forms are now a standard tool in the string-theoretic computation of black hole entropy. Review articles in theoretical physics journals cite Ramanujan's 1920 letter directly as a primary source. The applications have also flowed back into mathematics: new identities discovered from physics considerations have sharpened the theory of mock modular forms themselves. The feedback loop between Ramanujan's last letter and the mathematics of quantum gravity remains open and active.
An idea has no obligation to know what it is for. Ramanujan could not have anticipated black holes; the black hole physicists could not have anticipated a devotional autodidact in 1920 Kumbakonam. The deep work had to be done anyway, by both parties, on faith that somewhere down the line the connection would be made.
Ninety-plus years separate Ramanujan's 1920 letter from the 2012 Dabholkar-Murthy-Zagier application of mock modular forms to supersymmetric black hole entropy.
Historical context
Final year of Ramanujan's life and the eighty-year aftermath: late British India (1919 to 1920), twentieth-century Cambridge and Princeton mathematics, and the twenty-first-century reopening of the field
Ramanujan returned to India in March 1919 from Cambridge in poor health. He lived his final year between Kumbakonam, Kodumudi, and Madras, supported by his wife Janaki and his mother Komalatammal, and in contact with his friend R. Ramachandra Rao. He died in Kumbakonam on April 26, 1920. The India that received him back was on the brink of Gandhi's Non-Cooperation Movement, with the Jallianwala Bagh massacre still fresh from April 1919 and the Khilafat movement in motion. The final letter to Hardy on January 12, 1920, was written in this setting.
Living traditions
The SASTRA Ramanujan Award, given annually at Kumbakonam, recognizes outstanding mathematicians under 32 working in fields Ramanujan influenced; past winners include Manjul Bhargava, Terence Tao, Akshay Venkatesh, and Kannan Soundararajan. The International Centre for Theoretical Sciences in Bengaluru has hosted recurring programs on mock modular forms and black hole physics, where work on mock theta functions remains active. A 2016 feature film, The Man Who Knew Infinity, introduced Ramanujan's story, and the January 1920 letter to a global audience, and a 1988 NOVA documentary had already brought Namagiri and mock theta functions into mainstream American mathematical culture.
- National Mathematics Day Observance: Since 2012, December 22, Ramanujan's birthday, has been observed across India as National Mathematics Day, declared by Prime Minister Manmohan Singh during the 125th anniversary of Ramanujan's birth. Schools, universities, and research institutes hold lectures, problem-solving competitions, and readings from Ramanujan's life and work. The day has become a focal point for mathematical outreach to Indian schoolchildren.
- Namagiri Thayar Temple, Namakkal: The temple of Namagiri Thayar, the family deity of Ramanujan's household and the goddess he credited with giving him his mathematical visions. Namagiri is a form of Mahalakshmi, consort of Narasimha in the adjacent Namakkal Lakshmi Narasimha rock-cut temple. The two temples together form the spiritual geography that Ramanujan carried with him from South India to Cambridge and back.
- Ramanujan's House (SASTRA Ramanujan Museum): The house where Ramanujan grew up and where, during his final months in 1919 and 1920, he wrote the last letter to Hardy introducing mock theta functions. The property was acquired and restored by SASTRA University as a museum; visitors can see the small room where he worked, original manuscript facsimiles, and the staircase Janaki climbed to bring him food in his final weeks. The nearby Sarangapani Temple, where the family worshipped, is a classical Divya Desam of the Sri Vaishnava tradition.
Reflection
- Is there a piece of work you have held back because you cannot yet justify it to the standards your field demands?
- Why do you think Ramanujan, in his final months, spent his strength on inventing mock theta functions rather than on a clean summary of what he already understood?
- What is the relationship between intuition that runs ahead of proof and proof that runs ahead of intuition?