Journal of the Ramanujan Mathematical Society

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Generalization of an Identity of Ramanujan


Affiliations
1 Institute for Mathematical Sciences, C.I.T Campus, Taramani, Chennai-600113, India
2 Department of Mathematics, Queen’s University, Kingston, Ontario-K7L3N6, Canada
     

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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

∑anTn and ∑bnTn

which are both rational functions with certain property, we then explicitly show that

∑anbnTn

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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  • Generalization of an Identity of Ramanujan

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Authors

Sanoli Gun
Institute for Mathematical Sciences, C.I.T Campus, Taramani, Chennai-600113, India
M. Ram Murty
Department of Mathematics, Queen’s University, Kingston, Ontario-K7L3N6, Canada

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

∑anTn and ∑bnTn

which are both rational functions with certain property, we then explicitly show that

∑anbnTn

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.