Asymptotic formulae in the theory of partitions

Abstract

It is the object of this paper to obtain an asymptotic formula for the number of partitions pm(n) of a large positive integer n into m parts λr, where the number m becomes large with n and the numbers λ1, λ2,… form a sequence of positive integers. The formula is proved by using the classical method of contour integration due to Hardy, Ramanujan and Littlewood. It will be necessary to assume certain conditions on the sequence λr, but these conditions are satisfied in most of the cases of interest. In particular, we shall be able to prove the asymptotic formula in the cases of partitions into positive integers, primes and kth powers for any positive integer k.