Resolution of a conjecture of Andrews and Lewis involving cranks of partitions
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- by Daniel M. Kane PDF
- Proc. Amer. Math. Soc. 132 (2004), 2247-2256 Request permission
Abstract:
Andrews and Lewis have conjectured that the sign of the number of partitions of $n$ with crank congruent to 0 mod 3, minus the number of partitions of $n$ with crank congruent to 1 mod 3, is determined by the congruence class of $n$ mod 3 apart from a finite number of specific exceptions. We prove this by using the “circle method" to approximate the value of this difference to great enough accuracy to determine its sign for all sufficiently large $n$.References
- George A. Andrews and Richard Lewis, The ranks and cranks of partitions moduli 2, 3, and 4, J. Number Theory 85 (2000), no. 1, 74–84. MR 1800302, DOI 10.1006/jnth.2000.2537
- Tom M. Apostol, Modular functions and Dirichlet series in number theory, 2nd ed., Graduate Texts in Mathematics, vol. 41, Springer-Verlag, New York, 1990. MR 1027834, DOI 10.1007/978-1-4612-0999-7
- George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
Additional Information
- Daniel M. Kane
- Affiliation: 2814 Regent Street, Madison, Wisconsin 53705
- Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
- Email: dankane@mit.edu
- Received by editor(s): February 18, 2003
- Received by editor(s) in revised form: May 15, 2003
- Published electronically: March 3, 2004
- Communicated by: Wen-Ching Winnie Li
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 2247-2256
- MSC (2000): Primary 11P82, 11P83
- DOI: https://doi.org/10.1090/S0002-9939-04-07353-8
- MathSciNet review: 2052400