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Journal of the Ramanujan Mathematical Society

Volume 31, Issue 2, June 2016  pp. 125–135.

Generalization of an identity of Ramanujan

Authors:  Sanoli Gun and M. Ram Murty
Author institution:Institute for Mathematical Sciences, C.I.T Campus, Taramani, Chennai 600 113, India

Summary:  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 ∑ n=0 ∞ an Tn and ∑ n = 0 ∞ bn Tn which are both rational functions with certain property, we then explicitly show that ∑ n=0 ∞ an bn Tn 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.

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