## 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, a_{0}, and b_{0}. 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 language | English (US) |
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Pages (from-to) | 1277-1307 |

Number of pages | 31 |

Journal | Mathematische Zeitschrift |

Volume | 290 |

Issue number | 3-4 |

DOIs | |

State | Published - Dec 1 2018 |

## Keywords

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

## ASJC Scopus subject areas

- Mathematics(all)