On Ramanujan and Dirichlet Series with Euler Products

In his unpublished manuscripts (referred to by Birch [1] as Fragment V, pp. 247-249), Ramanujan [3] gave a whole list of assertions about various (transforms of) modular forms possessing naturally associated Euler products, in more or less the spirit of his extremely beautiful paper entitled "On certain arithmetical functions" (in Trans. Camb. Phil. Soc. 22 (1916)). It is simply amazing how Ramanujan could write down (with an ostensibly profound insight) a basis of eigenfunctions (of Hecke operators) whose associated Dirichlet series have Euler products, anticipating by two decades the famous work of Hecke and Petersson. That he had further realized, in the event of a modular form / not corresponding to an Euler product, the possibility of restoring the Euler product property to a suitable linear combination of modular forms of the same type as /, is evidently fantastic. Ramanujan's assertions referred to at the beginning were recently upheld and elucidated in the light of the work of Hecke, Rankin and Serre by Rangachari [4]. Page 5 of [4] refers to "an additional list of Euler product developments found in [3] which is incomplete except for the first one" (already covered by one of Ramanujan's abovementioned assertions): namely,