# Partitions into kth powers of terms in an arithmetic progression

Partitions into kth powers of terms in an arithmetic progression 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 $$k=2$$ k = 2 . In this paper, we consider partitions into parts from a specific set $$A_k(a_0,b_0) :=\left\{ m^k : m \in \mathbb {N} , m \equiv a_0 \,(\text {mod}\,b_0) \right\}$$ A k ( a 0 , b 0 ) : = m k : m ∈ N , m ≡ a 0 ( mod b 0 ) , for fixed positive integers k, $$a_0,$$ a 0 , and $$b_0$$ 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). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

# Partitions into kth powers of terms in an arithmetic progression

, Volume 290 (4) – May 29, 2018
31 pages

/lp/springer_journal/partitions-into-kth-powers-of-terms-in-an-arithmetic-progression-gTw9qtScTe
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
D.O.I.
10.1007/s00209-018-2063-8
Publisher site
See Article on Publisher Site

### 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 $$k=2$$ k = 2 . In this paper, we consider partitions into parts from a specific set $$A_k(a_0,b_0) :=\left\{ m^k : m \in \mathbb {N} , m \equiv a_0 \,(\text {mod}\,b_0) \right\}$$ A k ( a 0 , b 0 ) : = m k : m ∈ N , m ≡ a 0 ( mod b 0 ) , for fixed positive integers k, $$a_0,$$ a 0 , and $$b_0$$ 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).

### Journal

Mathematische ZeitschriftSpringer Journals

Published: May 29, 2018

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