Partitions into kth powers of terms in an arithmetic progression

Partitions into kth powers of terms in an arithmetic progression Math. Z. https://doi.org/10.1007/s00209-018-2063-8 Mathematische Zeitschrift Partitions into kth powers of terms in an arithmetic progression 1 1,3 Bruce C. Berndt · Amita Malik · 1,2 Alexandru Zaharescu Received: 26 June 2016 / Accepted: 9 March 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 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 A (a , b ) := k 0 0 m : m ∈ N, m ≡ a (mod b ) , for fixed positive integers k, a , and b .Wegiveanasymp- 0 0 0 0 totic 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). Keywords Partitions · Parity · Arithmetic progression http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Partitions into kth powers of terms in an arithmetic progression

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature
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

Math. Z. https://doi.org/10.1007/s00209-018-2063-8 Mathematische Zeitschrift Partitions into kth powers of terms in an arithmetic progression 1 1,3 Bruce C. Berndt · Amita Malik · 1,2 Alexandru Zaharescu Received: 26 June 2016 / Accepted: 9 March 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 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 A (a , b ) := k 0 0 m : m ∈ N, m ≡ a (mod b ) , for fixed positive integers k, a , and b .Wegiveanasymp- 0 0 0 0 totic 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). Keywords Partitions · Parity · Arithmetic progression

Journal

Mathematische ZeitschriftSpringer Journals

Published: May 29, 2018

References

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