Srinivasa Aiyangar Ramanujan

1. Ramanujan’s Series for \(\frac{1}{\pi}\)

Discovered in 1914, this formula stunned mathematicians and later became central to high-precision computations of \(\pi\).

$$ \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^{\infty} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}} $$

2. Ramanujan Tau Function

Introduced in 1916, this function linked modular forms and number theory in ways only fully understood decades later.

$$ \sum_{n=1}^{\infty} \tau(n) q^n = q \prod_{n=1}^{\infty} (1 - q^n)^{24} $$

3. Hardy–Ramanujan Partition Formula

Derived in 1918, this asymptotic formula revolutionized understanding of integer partitions using analytic methods.

$$ p(n) \sim \frac{1}{4n\sqrt{3}} e^{\pi\sqrt{\frac{2n}{3}}} $$

4. Ramanujan’s Continued Fractions

These formulas emerged from Ramanujan’s notebooks and baffled mathematicians until modern modular theory explained them.

$$ R(q) = q^{1/5}\prod_{n=1}^{\infty}\frac{(1-q^{5n-1})(1-q^{5n-4})}{(1-q^{5n-2})(1-q^{5n-3})} $$