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Proceedings of the American Mathematical Society Series B

ISSN 2330-1511

   
 
 

 

Disproof of a conjecture by Rademacher on partial fractions


Authors: Michael Drmota and Stefan Gerhold
Journal: Proc. Amer. Math. Soc. Ser. B 1 (2014), 121-134
MSC (2010): Primary 11P82, 41A60
DOI: https://doi.org/10.1090/S2330-1511-2014-00014-6
Published electronically: November 21, 2014
MathSciNet review: 3280294
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Abstract: In his book Topics in Analytic Number Theory (1973), Hans
Rademacher considered the generating function of integer partitions into at most $ N$ parts and conjectured certain limits for the coefficients of its partial fraction decomposition. We carry out an asymptotic analysis that disproves this conjecture, thus confirming recent observations of Sills and Zeilberger (Journal of Difference Equations and Applications 19 (2013)), who gave strong numerical evidence against the conjecture.


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Additional Information

Michael Drmota
Affiliation: Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstraße 8–10/105-1, A-1040 Vienna, Austria
Email: michael.drmota@tuwien.ac.at

Stefan Gerhold
Affiliation: Institute of Mathematical Methods in Economics, Vienna University of Technology, Wiedner Hauptstraße 8–10/105-1, A-1040 Vienna, Austria
Email: sgerhold@fam.tuwien.ac.at

DOI: https://doi.org/10.1090/S2330-1511-2014-00014-6
Keywords: Integer partitions, partial fraction decomposition, Mellin transform, polylogarithm, saddle point asymptotics
Received by editor(s): December 16, 2013
Received by editor(s) in revised form: June 12, 2014
Published electronically: November 21, 2014
Additional Notes: The authors gratefully acknowledge financial support from the Austrian Science Fund (FWF) under grants P 24880-N25 (second author), resp. F5002 (first author).
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 by the author under Creative Commons Attribution 3.0 License (CC BY 3.0)