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

DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month Explore the DeepDyve Library Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve Freelancer DeepDyve Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations