Generalization Of An Identity of Ramanujan

Abstract

In this article, we extend two identities proved by Ramanujan involving the Riemann zeta function and the Dirichlet L-function associated to the non-trivial Dirichlet character modulo 4. More precisely, given two power series Σ∞ n=0 anT n and Σ∞ n=0 bnT n which are both rational functions with certain property, we then explicitly show that Σ∞ n=0 an bnT n is again a rational function with the same property. We use this to explain Ramanujan’s identities and also analyse Rankin-Selberg convolutions of automorphic L-functions.