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The previous papers of this series are I. Plane partitions.Quart. J. of Math. (Oxford series), 2 (1931), 177–189. II. Weighted partitions.Proc. London Math. Soc. (2), 36 (1933), 117–141. The present paper may be read independently of these.
Proc. London Math. Soc. (2), 17 (1918), 75–115. The reader will find it interesting to compare the methods and results of this paper with my work here.
The problem was suggested to me by Professor Hardy, to whom my thanks are also due for much valuable advice in the course of the investigation.
Journal London Math. Soc., 8 (1933), 71–79.
Ifk=1, thenj=−1/24. Readers familiar with the memoir of Hardy and Ramanujan referred to above will recollect the appearance of the number −1/24.
Lemma 33.
Whenk=1, my definition of ωp, q differs in form from that appearing in the transformation used by these authors. The latter definition does not involve the sum\(\sum\limits_{h = 1}^{q - 1} {hdh}\) H. Rademacher (“Zur Theorie der Modulfunktionen”,Journal für Math., 167 (1932), 312–336) obtained the modular function transformation forf(x) whenk=I with the form of ωp, q that I use here. The same author showed (Journal London Math. Soc., 7 (1932), 14–19) by an elementary method that the two definitions are equivalent.
Ifq=1, thenp=o,Y=y,h=q, andS q =−logf′(x),S′ q =−logP 0,1, so that Theorem 4 follows at once from Lemma 5. This enables us to shorten our work considerably if our only object is the proof of Theorem 2 (see § 12.2).
Landau,Vorlesungen über Zahlentheorie, I, (Leipzig, 1927), Satz 267.
Landau,Vorlesungen I, Satz 267 (k=2) and Satz 315 (k≥3). Fork=I, the result is trivial.
Landau,Handbuch der Lehre von der Verteilung der Primzahlen I, 216–219.
Jour. London Math. Soc., 8 (1933), 71–79.
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Wright, E.M. Asymptotic partition formulae. III. Partitions intok-th powers. Acta Math. 63, 143–191 (1934). https://doi.org/10.1007/BF02547353
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DOI: https://doi.org/10.1007/BF02547353