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Zur Frage der Zerlegung von Zahlen in eine unbeschränkte Anzahl von Summanden. (Russian) Zbl 0057.03803

The author states two Tauberian theorems for partitions. One is of the strong type considered by A. E. Ingham [Ann. Math. (2) 42, 1075–1090 (1941; Zbl 0063.02973)], F. C. Auluck and C. B. Haselgrove [Proc. Camb. Philos. Soc. 48, 566–570 (1952; Zbl 0047.28002)], C. B. Haselgrove and H. N. V. Temperley [Proc. Camb. Philos. Soc. 50, 225–241 (1954; Zbl 0055.27401)], and G. Meinardus [Math. Z. 59, 388–398 (1954; Zbl 0055.03806)]. The other is identical with a weaker but more comprehensive result of K. Knopp [Schrift. Königsberg. gelehrt. Ges., naturwiss. Kl. 2, 45–74 (1925; JFM 51.0146.01)], which was subsequently proved again by N. A. Brigham in a more general form [Proc. Am. Math. Soc. 1, 182–191 (1950; Zbl 0037.16903)]. The author remarks that the converse of Knopp’s theorem is also true; in fact, this was proved by P. Erdős [Ann. Math. (2) 43, 437–450 (1942; Zbl 0061.07905)].
Reviewer: P. T. Bateman

MSC:

11P82 Analytic theory of partitions
40E05 Tauberian theorems
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