In his famous paper, «On certain arithmetical functions», Ramanujan offers for the first time the Euler product of the Dirichlet series in which the coefficients are given by Ramanujan's tau-function. In his lost notebook, Ramanujan records further Euler products for L-series attached to modular forms, and, typically, does not record proofs for these claims. In this semi-expository article, for the Euler products appearing in his lost notebook, we provide or sketch proofs using elementary methods, binary quadratic forms, and modular forms.

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Keywords binary quadratic forms, Dedekind eta-function, Dirichlet series with Euler products, Eisenstein series, Ramanujan tau-function, Ramanujan's lost notebook
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Journal International Journal of Number Theory
Berndt, B.C. (Bruce C.), Kim, B. (Byungchan), & Williams, K.S. (2013). Euler products in ramanujan's lost notebook. International Journal of Number Theory, 9(5), 1313–1349. doi:10.1142/S1793042113500292