Fourier coefficients of cusp forms

Abstract

The object of this survey article is to trace the influence on the theory of modular forms of the ideas contained in L. J. Mordell's important paper ‘On Mr Ramanujan's empirical expansions of modular functions’, which appeared in October 1917 in this Society's Proceedings [32]. The equally important paper [42] by S. Ramanujan, ‘On certain arithmetical functions’, referred to in Mordell's title, was published in May 1916 in the same Society's older journal, the Transactions, which was regrettably suppressed in 1928, 107 years after its foundation. Ramanujan's paper was concerned not only with multiplicative properties of Fourier coefficients of modular forms, but also with their order of magnitude. Since subsequent papers on the latter subject have also appeared in the Proceedings, it seems appropriate to include further developments in this field of study in the present survey.