Partitions into kth powers of terms in an arithmetic progression

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Abstract

G. H. Hardy and S. Ramanujan established an asymptotic formula for the number of unrestricted partitions of a positive integer, and claimed a similar asymptotic formula for the number of partitions into perfect kth powers, which was later proved by E. M. Wright. Recently, R. C. Vaughan provided a simpler asymptotic formula in the case k= 2. In this paper, we consider partitions into parts from a specific set Ak(a0,b0):={mk:m∈N,m≡a0(modb0)}, for fixed positive integers k, a0, and b0. We give an asymptotic formula for the number of such partitions, thus generalizing the results of Wright and Vaughan. Moreover, we prove that the number of such partitions is even (odd) infinitely often, which generalizes O. Kolberg’s theorem for the ordinary partition function p(n).

Original languageEnglish (US)
Pages (from-to)1277-1307
Number of pages31
JournalMathematische Zeitschrift
Volume290
Issue number3-4
DOIs
StatePublished - Dec 1 2018

Keywords

  • Arithmetic progression
  • Asymptotics
  • Hardy–Littlewood circle method
  • Parity
  • Partitions

ASJC Scopus subject areas

  • Mathematics(all)

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