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Concerning periodicity in the asymptotic behaviour of partition functions

Published online by Cambridge University Press:  09 April 2009

P. Erdös  and
B. Richmond
P. Erdös
Affiliation:
University of Waterloo University Avenue, Waterloo Ontario, Canada
B. Richmond
Affiliation:
University of Waterloo University Avenue, Waterloo Ontario, Canada
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Abstract

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Let PA(n) denote the number of partitions of n into summands chosen from the set A = {a1, a2, …}. De Bruijn has shown that in Mahler's partition problem (aν = rν) there is a periodic component in the asymptotic behaviour of PA(n). We show by example that this may happen for sequences that satisfy aν ν and consider an analogous phenomena for partitions into primes. We then consider corresponding results for partitions into distinct summands. Finally we obtain some weaker results using elementary methods.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 21 , Issue 4 , June 1976 , pp. 447 - 456
Copyright
Copyright © Australian Mathematical Society 1976

References

Mahler, K. (1940), ‘On a Special Functional Equation’, J. London Math. Soc. 15, 115–123.CrossRefGoogle Scholar
de Bruijn, N. G. (1943), ‘On Mahler's Partition Problem’, Indag. Math. 10, 210–220.Google Scholar
Pennington, W. B. (1953), ‘On Mahler's Partition Problem’, Annals of Math. (2) 57, 531–546.CrossRefGoogle Scholar
Schwarz, W. (1967), ‘C. Mahler's Partitions Problem’, J. reine angew. Math 229, 182–188.Google Scholar
Roth, K. F. and Szekeres, G. (1954), ‘Some Asymptotic formulae in the Theory of Partitions’, Quart. J. Math., Oxford (2), 244–259.Google Scholar
Richmond, L. B., ‘Moments of Partitions II’, Acta Arith. (to appear).Google Scholar
Tichmarsh, E. C. (1951), The Theory of the Riemann Zeta-function (Oxford, 1951).Google Scholar
Montgomery, H. L. (1971), Topics in Multiplicative Number Theory (Springer-Verlag, 1971).CrossRefGoogle Scholar
Prachar, K. (1957), Primzahlverteilung (Springer-Verlag, 1957).Google Scholar
Bateman, P. T. and Erdös, P. (1956), ‘Monotonicity of Partition Functions’, Mathematica 3, 1–14.Google Scholar
Erdös, P. (1943), ‘On an Elementary Proof of Some Asymptotic Formulas in the Theory of Partitions’, Annals of Math. (2) 43, 437–450.CrossRefGoogle Scholar
Newman, D. J. (1951), ‘The Evaluation of the Constant in the Formula for the Number of Partitions of n’, Amer. J. of Math., 73, 599–601.CrossRefGoogle Scholar