The structure of the number of representations function in a binary quadratic form

Abstract

This paper contains, primarily, the extension to any integral, binary quadratic form of the results of a recent articlet concerning positive, binary, quadratic forms. With suitable conventions almost all the results carry over without change, though some of the proofs need slight alterations. Incidentally, there are treated automorphs of binary quadratic forms, and (rather fully) properties of sets of representations (representations equivalent through automorphic transformations) in a binary quadratic form. 1. Dirichlett has already in all essentials extended the notion of number of representations to indefinite forms. We shall utilize the following equivalent definition. Two representations (x,y) and (x', y') of m in the formf = [a, b, c ],? that is, two integral solutions of (1) ax2+ bxy + cy2 = M,