Abstract
Ramanujan’s transformation formula for the Riemann zeta function at odd positive integers is a clear example of how special values, modular symmetry and quickly converging series come together. The formula doesn’t provide a simple arithmetic evaluation of the odd zeta values, but it explains them through a specific balance between two Lambert series with parameters that are reciprocal in a modular way. In this article we offer a complete proof of the formula using residue theory. Our method starts with a Mellin representation of the Lambert series. It moves the contour across its full set of poles, evaluates each residue explicitly and then applies the functional equation of the zeta function to identify the transformed integral. The calculation shows that the Bernoulli-number polynomial exactly matches the finite residue contribution. Meanwhile, the half-zeta normalization is determined by two special residues. A specific focus on the value at three demonstrates the formula's strong performance and includes a clear error bound. Final result shows a straightforward interpretation of Ramanujan’s identity as a reciprocity law for the Riemann zeta function.
