The Reversion of Class Number Relations and the Total Representation of Integers as Sums of Squares or Triangular Numbers

Abstract

'We shall discuss a set of new arithmetical functions defined in ?? 7, 8 relating to representations as sums of square or triangular numbers, the connection of these with class numbers, and means for calculating by recurrence the numerical values of the functions. The new functions first present themselves in reversing the class number formulas of the classical types due to Kronecker, Hermite and Liouville. They are themselves connected by many relations of a like simplicity, and seem to deserve attention on their own account. In section I we fix the notation and state the sense in which reversion is used throughout; in II the functions are defined and their generating series determined, the absolute convergence of these being proved incidentally; III contains four examples of the reversion of simple class number relations, and IV gives a short selection from the numerous recurrences between the functions, those chosen for presentation being among the most useful for numerical computations.