Srinivasa Ramanujan (1887–1920) was an Indian mathematical prodigy whose work on number theory, infinite series, and q-series left a legacy of “groundbreaking contributions”. Despite his short life, Ramanujan discovered deep results in partitions, modular forms, continued fractions and other areas.

Over a century later, his insights underpin a wide range of modern disciplines. For example, Ramanujan’s mathematics now appears in cutting-edge physics (string theory and black hole entropy), computer science (graph algorithms and expanders), cryptography (elliptic-curve systems and prime number methods), and data/signal analysis (periodic transforms). In each case, his original results – often encapsulated in q-series or theta-like functions – provide the analytical tools or the inspiration for new technologies.

1)Partition Theory and q-Series: Ramanujan’s asymptotic formula for the partition function $p(n)$ and his congruences mod 5, 7, 11 remain central in analytic number theory. His circle-method approach to partitions (the Hardy–Ramanujan formula) has been co-opted into string-theoretic counting via the and his generating-function identities now inform algorithms for computing partition counts efficiently.

2)Modular and Mock Modular Forms: Ramanujan’s discovery of new modular identities (including the tau function and Rogers–Ramanujan identities) has foreshadowed modern “moonshine” phenomena and conformal field theory. In physics, partition-generating functions that count states are often modular forms, and Ramanujan’s mock theta functions – mysterious in his time – reappeared in black hole microstate counting Freeman Dyson even predicted string theorists would extend their toolkit to include these mock-theta functions. a prediction borne out by recent work on AdS/CFT and black hole entropy.

3)Continued Fractions and Analytical Identities: Ramanujan’s myriad continued-fraction formulas (such as the Rogers–Ramanujan continued fraction) underlie many modern identities in q-series and hypergeometric functions. These continued fractions have been studied for efficient numerical approximation and for encoding symmetries in modular-type objects. For example, his rapidly convergent continued fractions are used in computing mathematical constants and in constructing special functions that appear in theoretical physics.

Few Contemporary examples include:

1)String Theory and Black Hole Physics: The degeneracy of black hole microstates in string theory is given by coefficients of modular and mock-modular forms. In the simplest models, the generating function of states is a modular form (much like Ramanujan’s partition function). Physicists use this fact to compute black hole entropy from first principles (via the Cardy formula, which is closely related to the Hardy–Ramanujan method). In more advanced work, correcting for multicenter black hole contributions produces “single-center” counts described by mock modular forms – precisely the type of functions Ramanujan introduced.

2)Moonshine and Conformal Field Theory: In mathematical “moonshine” theories, monstrous and umbral modules are encoded by q-series whose coefficients relate to Ramanujan’s mock theta functions. These surprising links have spawned a rich interplay between number theory and symmetry groups that was unimaginable in Ramanujan’s time

Throughout these applications, Ramanujan’s specific discoveries – partition formulas, theta- and mock-theta functions, class invariants and continued fractions – reappear in new guises. For instance, his partition congruences and asymptotic formulas underpin counting arguments in combinatorics and statistical physics. His modular equations are the ancestors of modern cryptographic elliptic-curve formulas. And his network of q-series identities has inspired algorithmic techniques in computational number theory (Johansson’s fast partition algorithms).