TY - JOUR

T1 - Partitions into kth powers of terms in an arithmetic progression

AU - Berndt, Bruce C.

AU - Malik, Amita

AU - Zaharescu, Alexandru

N1 - Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 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).

AB - 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).

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U2 - 10.1007/s00209-018-2063-8

DO - 10.1007/s00209-018-2063-8

M3 - Article

AN - SCOPUS:85047668080

SN - 0025-5874

VL - 290

SP - 1277

EP - 1307

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 3-4

ER -