Generalization of a Ramanujan identity

TL;DR

This paper explores whether Ramanujan's curious identity related to prime numbers of the form 4n+3 is unique or part of a broader class of similar identities, expanding understanding of Ramanujan's work.

Contribution

The work investigates the generalization of Ramanujan's identity, providing insights into its potential broader family of related identities involving prime numbers.

Findings

Identifies conditions under which similar identities may exist

Provides a framework for generating related identities

Suggests that Ramanujan's identity is part of a larger pattern

Abstract

The Euler product for the Landau--Ramanujan constant could have motivated a curious identity by Ramanujan that appears in his notebooks two times. This observation involves a square root and the first four prime numbers of the form $4n+3$, i.e., $3, 7, 11, 19$. Berndt asks whether Ramanujan's identity is an isolated result, or if there are other identities of this type. With this work we would like to give a possible answer to Berndt's question.