How did Srinivasa Ramanujan contribute to analytic number theory?

Srinivasa Ramanujan was a pivotal figure in 20th-century number theory, making profound contributions to analytic number theory despite his limited formal training.

His primary contributions include:

• Partition Functions: He pioneered the study of the partition of numbers, which calculates the number of ways a positive integer can be expressed as a sum of positive integers.12
• Prime Number Theory: Through his collaboration with Godfrey H. Hardy, he made significant advances in the study of prime numbers. His publications included notable works such as "Highly composite numbers" and a proof regarding the distribution of prime factors in numbers.23
• Mathematical Functions: Ramanujan developed theories on continued fractions, infinite series, and elliptic integrals. His work on "mock modular forms"—a concept he outlined in his final year—has since been verified and linked to modern phenomena like black holes.2

While his intuition was legendary, it is important to note that he initially lacked a formal understanding of mathematical proof, and some of his early theorems regarding prime numbers were incorrect.2