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 \(A_k(a_0,b_0) :=\left\{ m^k : m \in \mathbb {N} , m \equiv a_0 \,(\text {mod}\,b_0) \right\} \), for fixed positive integers k, \(a_0,\) and \(b_0\). 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).
Similar content being viewed by others
References
Ahlgren, S.: Distribution of parity of the partition function in arithmetic progressions. Indag. Math. (N.S.) 10, 173–181 (1999)
Apostol, T.M.: Introduction to Analytic Number Theory, vol. 1. Springer, Berlin (1976)
Berndt, B.C., Yee, A.J., Zaharescu, A.: New theorems on the parity of partition functions. J. Reine Angew. Math. 566, 91–109 (2004)
Davenport, H.: Multiplicative Number Theory, 3rd edn. Springer, New York (2000)
Fabrykowski, J., Subbarao, M.V.: Some new identities involving the partition function \(p(n)\). In: Mollin, R.A. (ed.) Number Theory, pp. 125–138. Walter de Gruyter, New York (1990)
Gafni, A.: Power partitions. J. Number Theory 163, 19–42 (2016)
Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. Lond. Math. Soc. 17, 75–115 (1918)
Kolberg, O.: Note on the parity of the partition function. Math. Scand. 7, 377–378 (1959)
Newman, M.: Periodicity modulo m and divisibility properties of the partition function. Trans. Am. Math. Soc. 97, 225–236 (1960)
Nicolas, J.-L.: Odd values of the partition function \(p(n)\). Int. J. Number Theory 2(4), 469–487 (2006)
Nicolas, J.-L., Ruzsa, I .Z., Sárközy, A.: On the parity of additive representation functions. With an appendix by J. P. Serre. J. Number Theory. 73, 292–317 (1998)
Nicolas, J.-L., Sárközy, A.: On the asymptotic behaviour of general partitions functions. Ramanujan J. 4(1), 29–39 (2000)
Nicolas, J.-L., Sárközy, A.: On the asymptotic behaviour of general partitions functions II. Ramanujan J. 7(1–3), 279–298 (2003)
Nörlund, N.E.: Vorlesungen über Differenzenrechnung. Chelsea, New York (1954)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Special Functions. Cambridge University Press, Cambridge (2010)
Ono, K.: Parity of the partition function in arithmetic progressions. J. Reine Angew. Math. 472, 1–15 (1996)
Ono, K.: The partition function in arithmetic progressions. Math. Ann. 312, 251–260 (1998)
Richmond, L.B.: A general asymptotic result for partitions. Can. J. Math. 27(5), 1083–1091 (1975)
Titchmarsh, E.C.: The Theory of the Riemann Zeta-function, 2nd edn. The Clarendon Press, Oxford University Press, New York (1986)
Vaughan, R.C.: The Hardy–Littlewood Method, 2nd edn. Cambridge University Press, Cambridge (1997)
Vaughan, R.C.: Squares: additive questions and partitions. Int. J. Number Theory 11(5), 1367–1409 (2015)
Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, reprint of the fourth (1927) edition. Cambridge University Press, Cambridge (1996)
Wright, E.M.: Asymptotic partition formulae, III, partitions into \(k\)th powers. Acta Math. 63, 143–191 (1934)
Yang, Y.: Partitions into primes. Trans. Am. Math. Soc. 352(6), 2581–2600 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Berndt, B.C., Malik, A. & Zaharescu, A. Partitions into kth powers of terms in an arithmetic progression. Math. Z. 290, 1277–1307 (2018). https://doi.org/10.1007/s00209-018-2063-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-018-2063-8