Analytic Theory of Partitions

Abstract

Preliminary remarks. The chapters of Part III dealt only with formal power series in which the indeterminate could not be replaced.¹ If we now, however, replace the indeterminate x by a complex variable x, the formal power series become power series in the ordinary sense to which the concept of convergence, totally absent in Part III, applies. The convergent power series are analytic functions, and the formal identities become equations between analytic functions. This step opens the whole store of analytic tools for the treatment of arithmetical problems in additive number theory.