Sums of two squares and the tau-function: Ramanujan's trail

TL;DR

This paper explores Ramanujan's historical claims regarding sums of two squares and the non-divisibility of the tau-function, highlighting his pioneering role in number theory and subsequent developments.

Contribution

It provides a detailed historical survey of Ramanujan's conjectures and discusses later mathematical progress related to his trail in number theory.

Findings

Ramanujan's estimates for sums of two squares

His conjectures on tau-function non-divisibility

Connections to later developments in number theory

Abstract

Ramanujan, in his famous first letter to Hardy, claimed a very precise estimate for the number of integers that can be written as a sum of two squares. Far less well-known is that he also made further claims of a similar nature for the non-divisibility of the Ramanujan tau-function for certain primes. In this survey, we provide more historical details and also discuss related later developments. These show that, as so often, Ramanujan was an explorer in a fascinating wilderness, leaving behind him a beckoning trail.