Cusp Forms of Given Level and Real Weight

For this Ramanujan Birth Centenary volume it seems appropriate to return to the paper [4], which is one of two papers written nearly 50 years ago under the general title Contributions to the study of Ramanujan's function τ(n) and similar arithmetical functions. It may be of historical interest to remark that this title was suggested to the author by his research supervisor, the late Professor Hardy; Hardy had a very strong interest in every aspect of Ramanujan’s work, although he was not particularly interested in the general theory of modular forms or functions. The paper, which will be referred to as R, was concerned with what would now be called cusp forms of integral weight k, level N and constant multiplier system, but it was noted on p. 358 (Remark B) that “ the results... hold, not only for modular forms, but also for any modular relative invariant”; this last term refers to what would now be called cusp forms of real weight k, level N and arbitrary non-constant multiplier systems. The generalization is not spelled out in detail in the paper, although there is a somewhat cryptic, and not entirely accurate, reference to it on the last page, which deals in a footnote with such forms of weight κ≤2/5.