Ramanujan’s Early Work on Continued Fractions
“I had never seen anything in the least like [it] before” — G.H. Hardy
On or about the 31st of January 1913, mathematician G.H. Hardy (1877-1947) of Trinity College at Cambridge University received a parcel of papers from Madras, India. The package included a cover letter where a young clerk by the name of Srinivasa Ramanujan (1887–1920) provided an introduction of himself and his precarious situation, as well as various mathematical claims about the domain of the gamma function and the distribution of prime numbers.
This essay provides a narration of Ramanujan and Hardy’s subsequent correspondence on the topic of continued fractions, in addition to an overview of the fractions Ramanujan claimed to have discovered, and their history. The essay is based in large part on the wonderful book Ramanujan: Letters and Commentary* by Berndt & Rankin (1991).