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A General Asymptotic Result for Partitions

Published online by Cambridge University Press:  20 November 2018

Bruce Richmond
Bruce Richmond*
Affiliation:
University of Manitoba, Winnipeg, Canada
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In this paper we are concerned with partition functions pϒ(n) that have generating functions of the form

where γ(n) ≧ 0. We shall obtain an asymptotic relation for pϒ(n) under suitable restrictions on ϒ (see Theorem 1.1). These restrictions are weaker than those of Brigham [2] who considered this problem previously.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 5 , 01 October 1975 , pp. 1083 - 1091
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Bateman, P. T. and P. Erdôs, Monotonicity of partition functions, Mathematics 3 (1956), 1–14.Google Scholar
2. Brigham, N. A., A general asymptotic formula for partition functions, Proc. Amer. Math. Soc. 1 (1950), 182–191.Google Scholar
3. Brigham, N. A., On a certain weighted partition function, Proc. Amer. Math. Soc. 1 (1950), 192–204.Google Scholar
4. Gordon, B., Asymptotic formulas for restricted plane partitions, Proc. Symp. Combinatorics A.M.S.Google Scholar
5. Gordon, B. and Houten, L., Notes on plane partitions, J. Combinatorial Theory 4 (1968), 72–99.Google Scholar
6. Prachar, K., Primzahlverteilung (Springer-Verlag, 1957).Google Scholar
7. Richmond, B., Asymptotic relations for partitions, J. Number Theory (to appear).Google Scholar
8. Richmond, B., Asymptotic relations for partitions, Trans. Amer. Math. Soc. (to appear).Google Scholar
9. Richmond, B., Asymptotic results for partitions (I) and a conjecture of Bateman and Erdôs, J. Number Theory (to appear).Google Scholar
10. The moments of partitions (II), Acta Arithmetica (to appear).Google Scholar
11. Roth, K. F. and Szekeres, G., Some asymptotic formulae in the theory of partitions, Quart. J. of Math. (Oxford) (2), 5 (1954), 241–259.Google Scholar
12. Titchmarsh, E. C., The theory of the Riemann zeta-function (Oxford, 1951).Google Scholar
13. Wright, E. M., Asymptotic partition formulae I, plane partitions, Quart. J. Math. (Oxford) (2), 2, 177–189.Google Scholar