Mathematical Proceedings of the Cambridge Philosophical Society

Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Asymptotic formulae in the theory of partitions

C. B. Haselgrovea1 and H. N. V. Temperleya1

a1 King's College Cambridge

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.

(Received September 10 1953)