Paper Details

Srinivasa Ramanujan (1887–1920), the self taught Indian mathematical genius, made extraordinary contributions to various branches of mathematics including number theory, infinite series, continued fractions, modular forms, elliptic functions, and special functions. Despite negligible formal training and a short life span, he compiled nearly 3,900 results—many identities and theorems—many without proofs, which later mathematicians have been engaged in proving and extending. This paper provides an analytical study of Ramanujan’s major contributions, situating them in the context of the mathematical knowledge of his time, reviewing later developments that stemmed from his work, analysing specific key contributions, and drawing conclusions about the lasting impact. It discusses the theoretical significance of his work (e.g., partition function asymptotics, mock theta functions, Ramanujan conjectures) as well as applications (in mathematical physics, computing, cryptography, etc.). The review of literature highlights how later scholars have built on Ramanujan’s notebooks, lost notebook, and his collaboration with G.H. Hardy, as well as how modern areas such as string theory, moonshine, and renormalization use or are inspired by Ramanujan’s results. Findings show that despite occasional lack of rigour or missing proof steps, the originality and depth of his results have had broad mathematical and even physical implications. The paper ends with suggestions for future research: more systematic study of unproved entries in his notebooks; improved exposition of his methods; exploring computational aspects of his series and modular forms; and using Ramanujan’s insights in emerging fields (quantum computing, complex networks).

Ramanujan; number theory; partition functions; mock theta functions; continued fractions; modular forms; infinite series; special functions.