“RAMANUJAN’S CONGRUENCES AND DYSON’S CRANK”
DOI:
https://doi.org/10.29121/granthaalayah.v2.i3.2014.3056Keywords:
Crank, Congruences, Eureka, ng Romantic Tale, Modulo, Theoretical, DescriptionAbstract [English]
In 1944, Freeman Dyson conjectured the existence of a “crank” function for partitions that would provide a combinatorial result of Ramanujan’s congruence modulo 11. In 1988, Andrews and Garvan stated such functions and described the celebrated result that the crank simultaneously explains the three Ramanujan congruences modulo 5, 7 and 11. Dyson wrote the article, titled Some Guesses in the theory of partitions, for Eureka, the undergraduate mathematics journal of Cambridge. He discovered the many conjectures in this article by attempting to find a combinatorial explanation of Ramanujan’s famous congruences for P (n), the number of partitions of n indeed, Ramanujan’s formulas lay unread until 1976 when Dyson found In the Trainty College Library of Cambridge University among papers from the estate of the late G.N.Watson. In 1986, F.Garvan wrote his Pennsylvania state Ph.D. Thesis Precisely on the formulas of Ramanujan relative to the crank. In view of this theoretical description, the story of the crank is a long romantic tale and the crank functions are intimately connected to all partitions congruences. In 2005, Mahlburg stated that the crank functions themselves obey Ramanujan type congruences.
Downloads
References
G.E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Application,
Vol. 2 (G.- C. Rota, ed.), Addison-Wesley, Reading, mass, 1976 (Reissued, Cambridge
University Press, London and New York,1985).
G.E. Andrews and F.G. Garvan, Dyson’s Crank of a Partition, Bulletin (New series) of the
American Mathematical Society, 18(2): 167–171. (1988). DOI: https://doi.org/10.15281/jplantres1887.2.18_167
A.O.L .Atkin and H.P.F. Swinnerton-Dyer, (1954), Some Properties of Partitions, Proc. DOI: https://doi.org/10.1112/plms/s3-4.1.84
London Math . Soc. 3(4): 84–106.
F. Dyson, Some guesses in the theory of partitions, (1944), Eureka ,Cambridge, U.K.
, 10-15.
F.G. Garvan, Ramanujan Revisited, Proceeding of the Centenary Conference, University of
Illinois, Urban-Champion.(1988).
F.G. Garvan, Dyson’s Rank Function and Andrews’ spt-function, University of Florida,
Seminar Paper Presented in the University of Newcastle on 20 August (2013).
F.G. Garvan, Generalizations of Dyson’s rank, Ph.D. Thesis, Pennsylvania State
University, (1986) .
. G.H. Hardy, E.M. Wright, (1965). Introduction to the Theory of Numbers, 4th Edition,
Oxford, Clarendon Press .
K. Mahlburg , Partitions congruences, (2005), Proc. Natl. Acad.Sci.USA102,15373- 15376. DOI: https://doi.org/10.1073/pnas.0506702102
Downloads
Published
How to Cite
Issue
Section
License
With the licence CC-BY, authors retain the copyright, allowing anyone to download, reuse, re-print, modify, distribute, and/or copy their contribution. The work must be properly attributed to its author.
It is not necessary to ask for further permission from the author or journal board.
This journal provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge.
