Ramanujan’s First Letter to G.H. Hardy (1913)

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 which included a cover letter from an aspiring young Indian mathematician by the name of Srinivasa Ramanujan (1887–1920). The cover letter discussed three topics:

1. An introduction of Ramanujan and his precarious situation;

2. A claim about the domain of the gamma function; and

3. A claim about the distribution of prime numbers;

This essay provides an overview of the mathematical content of Ramanujan’s first letter, as well as Hardy’s reaction and response.

The Cover Letter

Ramanujan’s letter may be divided into four paragraphs, covering essentially three topics: 1. An introduction of who he is; 2. A discussion of the gamma function for negative and fractional values; and 3. A discussion of the distribution of prime numbers.

The First Paragraph

Dear Sir,

I beg to introduce myself to you as a clerk in the Accounts Department of the Port Tru…