Euler products in ramanujan's lost notebook
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.
|Keywords||binary quadratic forms, Dedekind eta-function, Dirichlet series with Euler products, Eisenstein series, Ramanujan tau-function, Ramanujan's lost notebook|
|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