Almost-universal quadratic forms: An effective solution of a problem of Ramanujan

Abstract

In this article we investigate almost-universal positive-definite integral quaternary quadratic forms, that is, those representing sufficiently large positive integers. In particular, we provide an effective characterization of all such forms. In this way we obtain the final solution to a problem first addressed by Ramanujan in [12]. Special attention is given to $2$-anisotropic almost-universal quaternaries