The saddle-point method for general partition functions

Abstract

We apply the saddle-point method to derive asymptotic estimates or asymptotic series for the number of partitions of a natural integer into parts chosen from a subset of the positive integers whose associated Dirichlet series satisfies certain analytic properties. This enables grouping in a single statement many cases studied in the literature, as well as a number of new ones.

A few minor corrections with respect to the journal version are added in this version. In particular we corrected the application of the Phragmén-Lindelöf principle on p.3 which required a minor adaptation of the properties (d) and (f)