Mathematics of Computation

Author(s): Neil Calkin; Jimena Davis; Kevin James; Elizabeth Perez; Charles Swannack.
Journal: Math. Comp. 76 (2007), 1619-1638.
MSC (2000): Primary 05A17; Secondary 11P81, 11P83
Posted: February 28, 2007
Retrieve article in: PDF DVI PostScript

Abstract: In this paper we discuss efficient algorithms for computing the values of the partition function and implement these algorithms in order to conduct a numerical study of some conjectures related to the partition function. We present the distribution of

for $N \le 10^9$ for primes up to

and small powers of

and

1.: S. Ahlgren, Distribution of parity of the partition function in arithmetic progressions, Indag. Math. 10 (1999) 173-181. MR 1816213 (2002i:11102)
2.: S. Ahlgren, The partition function modulo composite integers , Math. Ann. 318 (2000), 795-803. MR 1802511 (2001j:11099)
3.: S. Ahlgren and M. Boylan Arithmetic properties of the partition function, Invent. Math. 153 (2003), no. 3, 487-502. MR 2000466 (2004e:11115)
4.: S. Ahlgren and M. Boylan Coefficients of half-integral weight modular forms modulo $l\sp j$ , Math. Ann. 331 (2005), no. 1, 219-239. MR 2107445 (2005k:11091)
5.: S. Ahlgren and M. Boylan Addendum: "Coefficients of half-integral weight modular forms modulo $l\sp j$ , Math. Ann. 331 (2005), no. 1, 219-239 MR 2107445 (2005k:11091)
6.: S. Ahlgren and K. Ono, Congruences and conjectures for the partition function, Contemporary Math., 291 (2001) 1-10. MR 1874518 (2002j:11120)
7.: S. Ahlgren and K. Ono, Congruence properties for the partition function, Proceedings of the National Academy of Sciences, 98 no. 9 (2001), 978-984. MR 1862931 (2002k:11187)
8.: G. E. Andrews, (1998) The theory of partitions, (1998) Cambridge University Press, Cambridge. MR 1634067 (99c:11126)
9.: A. O. L. Atkin, Multiplicative congruence properties and density problems for , Proc. London Math. Soc., 18 (1968) 563-576. MR 0227105 (37:2690)
10.: A. O. L. Atkin and J. N. O'Brien, Some properties of and modulo powers of 13, Trans. Amer. Math. Soc., 126 (1967) 442-459. MR 0214540 (35:5390)
11.: A. O. L. Atkin and H. P. F. Swinnerton-Dyer, Modular forms on noncongruence subgroups, Combinatorics, Proc. Sympos. Pure Math., UCLA, 1968, 19 (1971) 1-25. MR 0337781 (49:2550)
12.: J. Bruinier and K. Ono, Fourier coefficients of half-integral weight modular forms, J. Number Th. 99 (2003), 164-179. MR 1957250 (2004b:11056)
13.: J. Bruinier and K. Ono, Fourier coefficients of half-integral weight modular forms (corrigendum), J. Number Th. 104 (2004), 378-379. MR 2029514 (2004j:11044)
14.: Heath-Brown, D. R. Zero-free regions for Dirichlet -functions, and the least prime in an arithmetic progression. Proc. London Math. Soc. (3) 64 (1992), no. 2, 265-338. MR 1143227 (93a:11075)
15.: O. Kolberg, Note on the parity of the partition function, Math. Scand. 7 (1959), 377-378. MR 0117213 (22:7995)
16.: T. Klove, Recurrence formulae for the coefficients of modular forms and congruences for the partition function and for the coefficients of $j(\tau)$ , $(j(\tau)-1728)^{1/2}$ , and $j(\tau)^{1/3}$ , Math. Scand. 23 (1969), 133-159. MR 0252324 (40:5545)
17.: M. Newman, Periodicity modulo and divisibility properties of the partition function, Trans. Amer. Math. Soc. 97 (1960) 225-236. MR 0115981 (22:6778)
18.: J. -L. Nicolas, I. Z. Ruzsa and A. Sárközy (with an appendix by J. -P. Serre), On the parity of additive representation functions, J. number Theory, 73, (1998) 292-317. MR 1657968 (2000a:11151)
19.: K. Ono, Distribution of the partition function modulo , Ann. Math. 151 (2000) 293-307. MR 1745012 (2000k:11115)
20.: K. Ono, The partition function in arithmetic progressions, Math. Ann. 312 (1998) 251-260. MR 1671788 (2000j:11153)
21.: T. R. Parkin and D. Shanks, (1967) On the distribution of parity in the partition function, Math. Comp., 21, 466-480. MR 0227126 (37:2711)
22.: H. Rademacher, Topics in analytic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 169, Springer-Verlag, New York-Heidelberg, 1973. MR 0364103 (51:358)
23.: S. Ramunajan, Congruence properties of partitions, Proceedings of the London Mathematical Society, 19 (1919) 207-210.
24.: R. L. Weaver, New congruences for the partition function, Ramanujan J., 5 (2001), no. 1, 53-63. MR 1829808 (2002h:11102)
25.: N. Calkin, J. Davis, K. James, E. Perez, and C. Swannack, cited 2006: Statistics of the Integer Partition Function. Available online at http://allegro.mit.edu/~swannack/ Projects/partitionStats/index.html

Retrieve articles in Mathematics of Computation with MSC (2000): 05A17, 11P81, 11P83

Neil Calkin
Affiliation: Department of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634-0975
Email: calkin@clemson.edu

Jimena Davis
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695
Email: jldavis9@unity.ncsu.edu

Kevin James
Affiliation: Department of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634-0975
Email: kevja@clemson.edu

Elizabeth Perez
Affiliation: Applied Mathematics and Statistics, The Johns Hopkins University, G.W.C. Whiting School of Engineering, 302 Whitehead Hall, 3400 North Charles Street, Baltimore, Maryland 21218-2682
Email: eaperez@ams.jhu.edu

Charles Swannack
Affiliation: Department of Electrical and Computer Engineering, Clemson University, Clemson, South Carolina 29634
Address at time of publication: Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: swannack@mit.edu

DOI: 10.1090/S0025-5718-07-01966-7
PII: S 0025-5718(07)01966-7
Keywords: Partition function, discrete fast Fourier transforms
Received by editor(s): March 11, 2005
Received by editor(s) in revised form: July 10, 2006
Posted: February 28, 2007
Additional Notes: The authors were partially supported by NSF grant DMS-0139569
The third author was partially supported by NSF grant DMS-0090117
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS