Ramanujan's Formula for $\zeta(2n+1)$

TL;DR

This paper surveys Ramanujan's intriguing formula for odd zeta values, exploring its history, connection to modular forms, properties, and potential for further mathematical research.

Contribution

It provides a comprehensive overview of Ramanujan's formula for (2n+1), highlighting its significance, properties, and avenues for future study.

Findings

Discusses the historical development of Ramanujan's formula.

Explores the connection to modular forms and polynomial properties.

Identifies analogues, generalizations, and research opportunities.

Abstract

Ramanujan made many beautiful and elegant discoveries in his short life of 32 years, and one of them that has attracted the attention of several mathematicians over the years is his intriguing formula for $\zeta(2n+1)$. To be sure, Ramanujan's formula does not possess the elegance of Euler's formula for $\zeta(2n)$, nor does it provide direct arithmetical information. But, one of the goals of this survey is to convince readers that it is indeed a remarkable formula. In particular, we discuss the history of Ramanujan's formula, its connection to modular forms, as well as the remarkable properties of the associated polynomials. We also indicate analogues, generalizations and opportunities for further research.