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On Ingham's Tauberian theorem for partitions

Published online by Cambridge University Press:  24 October 2008

F. C. Auluck  and
C. B. Haselgrove
F. C. Auluck
Affiliation:
Delhi UniversityDelhi
C. B. Haselgrove
Affiliation:
King's CollegeCambridge
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Extract

The object of this paper is to extend the range of validity of Ingham's Tauberian theorem for partitions (2). Ingham's theorem deals with many types of partitions, but it cannot be used in some cases of importance. With a slight modification of Ingham's notation we define

where P(u) is the number of solutions of

in integers ni ≤ 0, and the λi are a given set of numbers, not necessarily integers, such that 0 < λ1 < λ2 <.… If N(u), the number of λi not exceeding u, is given by

where B > 0, β > 0, and as u → ∞

then, according to Ingham's theorem, when u → ∞

where

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 48 , Issue 4 , October 1952 , pp. 566 - 570
Copyright
Copyright © Cambridge Philosophical Society 1952

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References

REFERENCES

(1)Hardy, G. H.Divergent series (Oxford, 1949).Google Scholar
(2)Ingham, A. E.Ann. Math., Princeton, (2), 42 (1941), 1075.CrossRefGoogle Scholar
(3)Wright, E. M.Quart. J. Math. 2 (1931), 177.CrossRefGoogle Scholar