On the parity of the number of multiplicative partitions and related problems
HTML articles powered by AMS MathViewer
- by Paul Pollack
- Proc. Amer. Math. Soc. 140 (2012), 3793-3803
- DOI: https://doi.org/10.1090/S0002-9939-2012-11254-7
- Published electronically: March 15, 2012
- PDF | Request permission
Abstract:
Let $f(N)$ be the number of unordered factorizations of $N$, where a factorization is a way of writing $N$ as a product of integers all larger than $1$. For example, the factorizations of $30$ are \[ 2\cdot 3\cdot 5,\quad 5\cdot 6, \quad 3\cdot 10, \quad 2 \cdot 15,\quad 30, \] so that $f(30)=5$. The function $f(N)$, as a multiplicative analogue of the (additive) partition function $p(N)$, was first proposed by MacMahon, and its study was pursued by Oppenheim, Szekeres and Turán, and others.
Recently, Zaharescu and Zaki showed that $f(N)$ is even a positive proportion of the time and odd a positive proportion of the time. Here we show that for any arithmetic progression $a\operatorname {mod} m$, the set of $N$ for which \[ f(N) \equiv a( \operatorname {mod} m) \] possesses an asymptotic density. Moreover, the density is positive as long as there is at least one such $N$. For the case investigated by Zaharescu and Zaki, we show that $f$ is odd more than 50 percent of the time (in fact, about 57 percent).
References
- Scott Ahlgren and Matthew Boylan, Coefficients of half-integral weight modular forms modulo $l^j$, Math. Ann. 331 (2005), no. 1, 219–239. MR 2107445, DOI 10.1007/s00208-004-0555-9
- Scott Ahlgren and Ken Ono, Congruences and conjectures for the partition function, $q$-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000) Contemp. Math., vol. 291, Amer. Math. Soc., Providence, RI, 2001, pp. 1–10. MR 1874518, DOI 10.1090/conm/291/04889
- R. Balasubramanian and Florian Luca, On the number of factorizations of an integer, Integers 11 (2011), A12, 5. MR 2798647, DOI 10.1515/integ.2011.012
- Neil Calkin, Jimena Davis, Kevin James, Elizabeth Perez, and Charles Swannack, Computing the integer partition function, Math. Comp. 76 (2007), no. 259, 1619–1638. MR 2299791, DOI 10.1090/S0025-5718-07-01966-7
- E. R. Canfield, Paul Erdős, and Carl Pomerance, On a problem of Oppenheim concerning “factorisatio numerorum”, J. Number Theory 17 (1983), no. 1, 1–28. MR 712964, DOI 10.1016/0022-314X(83)90002-1
- Louis Comtet, Advanced combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR 0460128
- Michael Coons and Sander R. Dahmen, On the residue class distribution of the number of prime divisors of an integer, Nagoya Math. J. 202 (2011), 15–22. MR 2804543, DOI 10.1215/00277630-1260423
- M. Deléglise, M. O. Hernane, and J.-L. Nicolas, Grandes valeurs et nombres champions de la fonction arithmétique de Kalmár, J. Number Theory 128 (2008), no. 6, 1676–1716 (French, with English summary). MR 2419188, DOI 10.1016/j.jnt.2007.07.003
- G. Halász, On the distribution of additive and the mean values of multiplicative arithmetic functions, Studia Sci. Math. Hungar. 6 (1971), 211–233. MR 319930
- R. R. Hall, A sharp inequality of Halász type for the mean value of a multiplicative arithmetic function, Mathematika 42 (1995), no. 1, 144–157. MR 1346679, DOI 10.1112/S0025579300011426
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 6th ed., Oxford University Press, Oxford, 2008. Revised by D. R. Heath-Brown and J. H. Silverman; With a foreword by Andrew Wiles. MR 2445243
- L. Kalmár, A “factorisatio numerorum” problémájáról, Mat. Fiz. Lapok 38 (1931), 1–15.
- —, Über die mittlere Anzahl der Produktdarstellungen der Zahlen (Erste Mitteilung), Acta Litt. Sci. Szeged 5 (1931), 95–107.
- Martin Klazar and Florian Luca, On the maximal order of numbers in the “factorisatio numerorum” problem, J. Number Theory 124 (2007), no. 2, 470–490. MR 2321375, DOI 10.1016/j.jnt.2006.10.003
- Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen. 2 Bände, Chelsea Publishing Co., New York, 1953 (German). 2d ed; With an appendix by Paul T. Bateman. MR 0068565
- Florian Luca, Anirban Mukhopadhyay, and Kotyada Srinivas, Some results on Oppenheim’s “factorisatio numerorum” function, Acta Arith. 142 (2010), no. 1, 41–50. MR 2601047, DOI 10.4064/aa142-1-3
- W. F. Lunnon, P. A. B. Pleasants, and N. M. Stephens, Arithmetic properties of Bell numbers to a composite modulus. I, Acta Arith. 35 (1979), no. 1, 1–16. MR 536875, DOI 10.4064/aa-35-1-1-16
- P. A. MacMahon, Memoir on the theory of the compositions of numbers, Philos. Trans. Roy. Soc. London 184 (1893), 835–901.
- —, Dirichlet series and the theory of partitions, Proc. London Math. Soc. 22 (1923), 404–411.
- Abdelhak Mazouz, Analyse $p$-adique et nombres de Bell à deux variables, Bull. Belg. Math. Soc. Simon Stevin 3 (1996), no. 4, 377–390 (French). MR 1418935
- Morris Newman, Periodicity modulo $m$ and divisibility properties of the partition function, Trans. Amer. Math. Soc. 97 (1960), 225–236. MR 115981, DOI 10.1090/S0002-9947-1960-0115981-2
- Jean-Louis Nicolas, Parité des valeurs de la fonction de partition $p(n)$ et anatomie des entiers, Anatomy of integers, CRM Proc. Lecture Notes, vol. 46, Amer. Math. Soc., Providence, RI, 2008, pp. 97–113 (French, with English summary). MR 2437968, DOI 10.1090/crmp/046/07
- A. Oppenheim, On an arithmetic function, J. London Math. Soc. 1 (1926), 205–211, part II in 2 (1927), 123–130.
- Thomas R. Parkin and Daniel Shanks, On the distribution of parity in the partition function, Math. Comp. 21 (1967), 466–480. MR 227126, DOI 10.1090/S0025-5718-1967-0227126-9
- Chr. Radoux, Arithmétique des nombres de Bell et analyse $p$-adique, Bull. Soc. Math. Belg. 29 (1977), no. 1, 13–28 (French). MR 485676
- Georg J. Rieger, Sur les nombres de Cullen, Séminaire de Théorie des Nombres (1976–1977), CNRS, Talence, 1977, pp. Exp. No. 16, 9 (French). MR 509629
- W. Sierpiński, Elementary theory of numbers, 2nd ed., North-Holland Mathematical Library, vol. 31, North-Holland Publishing Co., Amsterdam; PWN—Polish Scientific Publishers, Warsaw, 1988. Edited and with a preface by Andrzej Schinzel. MR 930670
- G. Szekeres and P. Turán, Über das zweite Hauptproblem der “Factorisatio Numerorum”, Acta Litt. Sci. Szeged 6 (1933), 143–154.
- G. T. Williams, Numbers generated by the function $e^{e^{x}-1}$, Amer. Math. Monthly 52 (1945), 323–327. MR 12612, DOI 10.2307/2305292
- Alexandru Zaharescu and Mohammad Zaki, On the parity of the number of multiplicative partitions, Acta Arith. 145 (2010), no. 3, 221–232. MR 2733086, DOI 10.4064/aa145-3-2
Bibliographic Information
- Paul Pollack
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
- Address at time of publication: Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2
- MR Author ID: 830585
- Email: pppollac@illinois.edu, twonth@gmail.com
- Received by editor(s): May 4, 2011
- Published electronically: March 15, 2012
- Additional Notes: The author is supported by NSF award DMS-0802970.
- Communicated by: Ken Ono
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 3793-3803
- MSC (2010): Primary 11N64; Secondary 11P83, 11B73
- DOI: https://doi.org/10.1090/S0002-9939-2012-11254-7
- MathSciNet review: 2944720