Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-27T00:11:16.458Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Auluck, F. C., Chowla, S. and Gupta, H.J. Indian math. (Cl.) Soc., N.S., 6 (1942), 105.Google Scholar
(2)Auluck, F. C. and Haselgrove, C. B.Proc. Camb. phil. Soc. 48 (1952), 566.CrossRefGoogle Scholar
(3)Brigham, N. A.Proc. Amer. math. Soc. 1 (1950), 182, 192.CrossRefGoogle Scholar
(4)Erdös, P. and Lehner, J.Duke math. J. 8 (1941), 335.CrossRefGoogle Scholar
(5)Ingham, A. E.Ann. Math., Princeton, 42 (1941), 1075.CrossRefGoogle Scholar
(6)Landau, E.Vorlesungen über Zahlentheorie, vol. 1, VI (Leipzig, 1927), p. 248.Google Scholar
(7)Szekeres, G.Quart. J. Math. (2), 2 (1951), 85.CrossRefGoogle Scholar
(8)Tchudakoff, N. G.Ann. Math., Princeton, 48 (1947), 515.CrossRefGoogle Scholar
(9)Titchmarsh, E. C.Theory of Functions (2nd ed.) (Oxford, 1939).Google Scholar
(10)Wright, E. M.Acta Math., Stockh., 63 (1934), 143.CrossRefGoogle Scholar