What did Ramanujan discover?

Srinivasa Ramanujan (1887-1920) was an Indian mathematician who made extraordinary and highly original contributions to mathematics despite having little formal training in the subject. His work primarily focused on:

• Number Theory: This was his main area. He made significant discoveries related to:
• Partitions of numbers: The number of ways an integer can be written as a sum of positive integers. He developed the Hardy-Ramanujan asymptotic formula for the partition function.
• Highly composite numbers: Numbers with more divisors than any smaller positive integer.
• Properties of prime numbers: Including work related to the prime number theorem.
• Infinite Series and Continued Fractions: Ramanujan had an incredible intuition for infinite series and continued fractions, discovering thousands of identities and formulas. Some of his series for pi (π) are among the fastest converging series known.
• Mock Theta Functions: These were functions Ramanujan described in his last letters to G.H. Hardy. Their precise nature remained a mystery for decades but have since become an important area of research, connecting to various fields like modular forms and physics.
• Hypergeometric Series and Elliptic Integrals: He also made significant contributions to these areas.
• Ramanujan's Notebooks: Much of his work was recorded in notebooks without formal proofs. Proving his results and exploring the ideas within these notebooks has been a major mathematical endeavor for decades.
• Key Takeaway: Ramanujan made profound and often surprising discoveries in number theory, infinite series, continued fractions, and mock theta functions. His work opened up new fields of mathematical research and continues to inspire mathematicians today.