Generalization of a Ramanujan Identity

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.

• MSC: Primary 11A67
• Secondary 11E25

Acknowledgments

The author would like to thank the referees for their valuable suggestions, which greatly improved the presentation of the paper. The comments of Sándor Bozóki (MTA SZTAKI) and Máté Horváth (CrySyS Lab) are highly appreciated.