A Proof of Ramanujan's Classic $π$ Formula
Thang Pang Ern, Devandhira Wijaya Wangsa
TL;DR
The paper delivers a full proof of Ramanujan's classic π formula by weaving together elliptic integrals, theta functions, and hypergeometric series within the Ramanujan g-invariant framework. It leverages Zucker–Robertson lattice sums and Dirichlet–Edwards number-theoretic results to evaluate associated L-series, enabling an explicit computation of the g-invariant at $n=58$. A key achievement is the explicit computation $g_{58}= olinebreak frac{1}{2}igl(5+ oot 2race{29}igr)^{1/2}$ (so $g_{58}^2= rac{5+ oot 2race{29}}{2}$) and the derivation of $L_{-8}(1)$ and $L_{29}(1)$, which together yield the rapid-convergence Ramanujan-type series for $1/ pi$. By connecting modular data, Pell-type equations, and class-number theory, the work deepens the arithmetic understanding of Ramanujan-type π-series and their computational prowess.
Abstract
In 1914, Ramanujan presented a collection of 17 elegant and rapidly converging formulae for $π$. Among these, one of the most celebrated is the following series: \[\frac{1}π=\frac{2\sqrt{2}}{9801}\sum_{n=0}^{\infty}\frac{26390n+1103}{\left(n!\right)^4}\cdot \frac{\left(4n\right)!}{396^{4n}}\] In this paper, we give a full proof of this classic formula using hypergeometric series and a special type of lattice sums due to Zucker and Robertson. We will also use some results by Dirichlet and Edwards in algebraic number theory.