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Asymptotic formulae in the theory of partitions

Published online by Cambridge University Press:  24 October 2008

C. B. Haselgrove  and
H. N. V. Temperley
C. B. Haselgrove
Affiliation:
King's CollegeCambridge
H. N. V. Temperley
Affiliation:
King's CollegeCambridge
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Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 50 , Issue 2 , April 1954 , pp. 225 - 241
Copyright
Copyright © Cambridge Philosophical Society 1954

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References

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