A transformation formula involving the Gamma and Riemann zeta functions in Ramanujan's Lost Notebook

TL;DR

This paper proves a transformation formula involving the Gamma and Riemann zeta functions from Ramanujan's Lost Notebook, revealing a modular relation and providing multiple proofs of the key identity.

Contribution

It presents a new proof of Ramanujan's transformation formula, connecting Gamma and Riemann zeta functions, with implications for modular relations.

Findings

Established a new proof of Ramanujan's transformation formula

Derived a modular relation from the formula

Connected the formula to classical functions in number theory

Abstract

In 'The Lost Notebook and Other Unpublished Papers' of Ramanujan are present some manuscripts of Ramanujan in the handwriting of G. N. Watson which are 'copied from loose papers'. We present a proof of a beautiful formula of Ramanujan in one of these manuscripts, namely a transformation formula involving the Gamma function and Riemann Zeta function. This formula elegantly yields a modular relation. Later we also give another proof of the first equality of this transformation formula along the lines suggested by Andrew P. Guinand.