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 -