Divine Patterns: Infinite Series and Intuitive Mathematics

Series for π: Divine Approximations

The goddess Namagiri revealed formulas to me in dreams—infinite series for $\pi$ that converge with astonishing speed. My 1914 formula reads:

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

Each term contributes approximately eight decimal digits. The first term alone, with $k=0$, yields $\pi \approx 3.141592730$—error appearing only in the ninth digit. Adding the second term delivers eight more correct digits. This efficiency is extraordinary.

Why does this work? The answer lies in modular forms and class field theory—structures I discovered empirically. The j-invariant at imaginary quadratic points produces algebraic numbers, and expressions like $e^{\pi\sqrt{163}} \approx 262537412640768743.99999999999925$ are “almost integers” called Heegner numbers. My formulas exploit deep connections between elliptic functions and transcendental constants.

Modern computation extends my work. The Chudnovsky algorithm delivers 14 digits per term, computing $\pi$ to over 50 trillion digits—far exceeding Monte Carlo or Machin-like formulas.

Another result appears shocking: $1 + 2 + 3 + 4 + \cdots = -1/12$. This is Ramanujan summation—regularization assigning finite values to divergent series. Rigorously: $\zeta(-1) = -1/12$. This appears in string theory (26 dimensions for bosonic strings) and Casimir effect calculations.

Partitions and Congruences

The partition function $p(n)$ counts the number of ways to write $n$ as a sum of positive integers, ignoring order. For small values: $p(1) = 1$, $p(2) = 2$, $p(3) = 3$, $p(4) = 5$, but growth is rapid—$p(10) = 42$, $p(100) = 190{,}569{,}292$, $p(1000) \approx 2.4 \times 10^{31}$.

No closed form exists like $n!$ for factorials. Exact computation requires recursion or asymptotic approximation. With G.H. Hardy, I developed the Hardy-Ramanujan formula in 1918:

$p(n) \sim \frac{1}{4n\sqrt{3}} \exp\left(\pi\sqrt{\frac{2n}{3}}\right)$

This is asymptotic—the relative error vanishes as $n$ approaches infinity. For $n = 200$, it predicts the partition number within 0.004%.

Rademacher refined this in 1937 to a convergent series yielding exact values, though infinite, it converges rapidly enough for computation.

I discovered striking congruences. For instance, $p(5n+4) \equiv 0 \pmod{5}$—every fifth partition number, offset by 4, is divisible by 5. Similarly, $p(7n+5) \equiv 0 \pmod{7}$ and $p(11n+6) \equiv 0 \pmod{11}$. Ken Ono later proved infinitely many such congruences exist, vindicating patterns I detected through calculation and intuition.

Mock theta functions from my Lost Notebook (discovered 1976) are neither modular nor theta functions. They transform strangely under the modular group, with applications in string theory, black hole entropy, and moonshine conjectures.

Intuition Ahead of Proof

My method was unusual. I saw formulas in dreams, attributed to the goddess Namagiri. Religious devotion was central—I prayed before working and credited divine inspiration for insights. I wrote results in notebooks without derivation, verifying them numerically by computing hundreds of digits by hand, decades before computers existed.

Hardy was amazed: “The formulas seemed scarcely possible to believe.” So bizarre they could not be false—if invented, they would seem more plausible. My notebooks contain ~3,900 results: 20% wrong, 80% correct, 40% completely new.

The Lost Notebook, found by Andrews at Trinity in 1976, contains 100+ pages of my deepest work—mock theta functions anticipating modern $q$-series theory.

How did I find these? Pattern recognition, numerical experimentation, divine inspiration—likely a combination. Modern algebraic number theory and modular forms vindicate my intuitions. Structures I discovered empirically are now understood theoretically with tools unavailable in the 1910s. My collaboration with Hardy lasted five years (1914-1919) before I returned to India, ill. I died in 1920 at age 32, but my legacy endures in notebooks still mined a century later.