In the spring of 1976, the American mathematician George Andrews was visiting Trinity College, Cambridge, when he was directed to a box of papers in the college library. The box contained manuscripts that had belonged to G.N. Watson, a mathematician who had died in 1965. Among Watson’s papers, Andrews found 138 pages of mathematical formulas written in a distinctive hand. The pages were unnumbered, unorganized, and covered with results stated without proof. They were the work of Srinivasa Ramanujan, written during the last year of his life, and they had been sitting in the box, essentially forgotten, for over fifty years.
Andrews recognized immediately that he was looking at something extraordinary. The pages contained thousands of formulas, many of them new, many of them deep, and many of them far ahead of their time. The mathematical community would later call these pages the “lost notebook,” and their discovery launched a new era of Ramanujan scholarship that continues to this day.
How the Notebook Was Lost
Ramanujan died in Madras (now Chennai) on April 26, 1920, at the age of thirty-two. In the last months of his life, despite severe illness, he continued to work with extraordinary intensity. He wrote mathematical formulas on loose sheets of paper, recording results as they came to him, with little regard for order or completeness.
After his death, his widow Janaki Ammal sent his papers to the University of Madras, which forwarded them to G.H. Hardy at Cambridge. Hardy, Ramanujan’s mentor and collaborator, received the papers but apparently did not study them systematically. At some point, the papers were passed to G.N. Watson, who was compiling Ramanujan’s unpublished work. Watson studied some of the results but did not publish a comprehensive analysis. When Watson died in 1965, the papers were placed in storage at Trinity College.
For eleven years, they sat in a box, uncatalogued. It was only Andrews’s visit in 1976, motivated by his research on partitions and q-series (areas in which Ramanujan had made fundamental contributions), that brought them back to light.
What the Notebook Contains
The lost notebook is not a notebook in the conventional sense. It is a collection of loose sheets, written on both sides, containing formulas, identities, and infinite series with almost no explanation, proof, or context. Ramanujan wrote down results as he discovered them, often in a compressed notation that only he fully understood. Deciphering and proving these results has occupied mathematicians for nearly fifty years.
The major themes of the lost notebook include:
- Mock theta functions: These are the most important objects in the notebook. Ramanujan mentioned them in his last letter to Hardy (January 1920) but gave only examples and no general definition. The lost notebook contains extensive calculations related to mock theta functions, which have since been recognized as a major new class of mathematical objects with deep connections to number theory, combinatorics, and mathematical physics.
- q-series and partitions: Ramanujan continued his lifelong fascination with the theory of partitions (the number of ways an integer can be expressed as a sum of positive integers). The notebook contains new partition identities and new formulas for partition-generating functions.
- Continued fractions: The notebook contains remarkable continued fraction identities, many of which have been proved only recently using techniques unavailable in Ramanujan’s time.
- Theta functions and modular forms: Results connecting classical theta functions with modular forms, a topic that has since become central to modern number theory.
- Definite integrals: Exact evaluations of integrals that seem intractable, expressed in closed form with an elegance that is characteristic of Ramanujan’s style.
Mock Theta Functions: A New Continent
The most significant mathematical objects in the lost notebook are the mock theta functions. In his last letter to Hardy, Ramanujan described them as a new class of functions that resembled theta functions (well-studied objects in number theory) but lacked the modular transformation properties that define them. He listed seventeen examples and hinted at their properties but provided no general theory.
For decades, mock theta functions remained mysterious. Mathematicians could verify Ramanujan’s specific formulas but could not understand what mock theta functions were in a conceptual sense. It was as if Ramanujan had discovered a new continent but drawn only scattered maps of its coastline.
The breakthrough came in 2002, when Sander Zwegers, a Dutch doctoral student, showed that Ramanujan’s mock theta functions are the “holomorphic parts” of certain harmonic Maass forms, objects that generalize the classical modular forms central to modern number theory. This placed mock theta functions in a precise mathematical framework, connecting them to the deepest structures in number theory, algebraic geometry, and mathematical physics.
Since Zwegers’s work, mock theta functions have found applications in string theory, statistical mechanics, the theory of black holes, and the calculation of quantum invariants of three-dimensional manifolds. Ramanujan, working on loose sheets of paper while dying of a disease that was consuming his body, had discovered mathematical objects that would prove essential to theoretical physics a century later.
Andrews and Berndt: The Systematic Study
After his discovery, George Andrews dedicated much of his career to studying the lost notebook. Together with Bruce Berndt, who had already spent decades editing Ramanujan’s other notebooks, Andrews undertook a systematic analysis and proof of every result in the lost notebook.
The project resulted in a five-volume series, Ramanujan’s Lost Notebook and Other Unpublished Papers, published between 2005 and 2018. The series runs to over 2,000 pages and contains proofs of thousands of Ramanujan’s formulas, many requiring sophisticated modern techniques that Ramanujan himself could not have used.
This raises a question that every mathematician who studies Ramanujan has asked: how did he find these results? Ramanujan rarely recorded proofs, and many of his formulas seem to come from nowhere, with no visible chain of reasoning connecting them to known results. Hardy described Ramanujan’s formulas as “a succession of beautiful and unexpected results, with no visible derivation.” Whether Ramanujan had proofs that he simply did not write down, or whether he arrived at his results by intuition and verification (computing special cases until the pattern became clear), remains one of the great mysteries of mathematical history.
The Other Notebooks
The lost notebook is not Ramanujan’s only manuscript legacy. His three earlier notebooks, compiled before and during his time at Cambridge (1914-1919), contain approximately 3,254 entries, most stated without proof. These notebooks were edited and annotated by Bruce Berndt over a period of two decades, resulting in a five-volume series published between 1985 and 1998.
Together, the notebooks and the lost notebook constitute one of the most extraordinary mathematical manuscripts in history. They contain results that span number theory, analysis, continued fractions, infinite series, modular forms, and special functions, recorded by a largely self-taught mathematician working in isolation (in India) and in collaboration (at Cambridge), often while seriously ill.
The sheer volume of unproved results has generated a small industry of Ramanujan scholarship. Proving a single entry from Ramanujan’s notebooks can be the basis of a published research paper or even a doctoral thesis. Many entries remain unproved decades after they were first studied. A few have been shown to be false, but the overwhelming majority have been confirmed as correct, a testament to the reliability of Ramanujan’s mathematical intuition.
A Legacy Still Unfolding
Srinivasa Ramanujan left behind a body of work that continues to generate new mathematics over a century after his death. The lost notebook, found by chance in a box at Trinity College, proved to be not an endpoint but a beginning: a source of problems, methods, and ideas that have enriched number theory, combinatorics, and mathematical physics in ways that Ramanujan himself could not have foreseen.
The story of the lost notebook is also a story about the fragility of mathematical knowledge. If Andrews had not visited Trinity College in 1976, or if Watson’s papers had been discarded rather than stored, Ramanujan’s final mathematical thoughts might have been lost forever. The mathematics would eventually have been rediscovered by others (mathematical truths, unlike physical artifacts, can always be found again), but the unique perspective and style that Ramanujan brought to these problems would have been gone.
That perspective, intuitive, visual, driven by pattern recognition rather than formal proof, remains one of the most remarkable in the history of mathematics. Ramanujan saw relationships between numbers that no one else could see. He recorded them on loose sheets of paper. He died before he could explain them. And then, fifty-six years later, a mathematician opened a box in a library and found them waiting.
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