Partition values and central critical values of certain modular -functions

Author:
John J. Webb

Journal:
Proc. Amer. Math. Soc. **138** (2010), 1263-1272

MSC (2010):
Primary 11F33; Secondary 11F11

DOI:
https://doi.org/10.1090/S0002-9939-09-10188-0

Published electronically:
December 8, 2009

MathSciNet review:
2578520

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the number of partitions of a positive number , let and let be the least non-negative residue of modulo . In this paper we prove congruences modulo between and ratios of central critical values of -functions associated to twists of certain integer weight newforms. In 1999, Guo and Ono proved analogous results for .

**[Ahl00]**S. Ahlgren,*Distribution of the partition function modulo composite integers*. Math. Ann. 318 (2000), no. 4, 795-803. MR**1802511 (2001j:11099)****[AB03]**S. Ahlgren and M. Boylan,*Arithmetic properties of the partition function*. Invent. Math. 153 (2003), no. 3, 487-502. MR**2000466 (2004e:11115)****[AO01]**S. Ahlgren and K. Ono,*Congruence properties for the partition function*. Proc. Natl. Acad. Sci. USA 98 (2001), no. 23, 12882-12884. MR**1862931 (2002k:11187)****[And98]**G. E. Andrews,*The Theory of Partitions*. Cambridge University Press, Cambridge, 1998. MR**1634067 (99c:11126)****[AL78]**A.O.L. Atkin and W.C.W. Li,*Twists of newforms and pseudo-eigenvalues of -operators*. Invent. Math. 48 (1978), no. 3, 221-243. MR**508986 (80a:10040)****[Gar07]**F.G. Garvan,*Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank*. Int. J. Number Theory, to appear. arXiv:0710.5793**[GO99]**L. Guo and K. Ono,*The partition function and the arithmetic of certain modular -functions*. Internat. Math. Res. Notices 1999, no. 21, 1179-1197. MR**1728677 (2000m:11102)****[Iwa97]**H. Iwaniec,*Topics in Classical Automorphic Forms*. Graduate Studies in Mathematics, 17. American Mathematical Society, Providence, RI, 1997. MR**1474964 (98e:11051)****[Kno93]**M. Knopp,*Modular Functions in Analytic Number Theory*. Chelsea Publishing Company, New York, 1993. MR**0265287 (42:198)****[Ono00]**K. Ono,*Distribution of the partition function modulo*. Ann. of Math. (2) 151 (2000), no. 1, 293-307. MR**1745012 (2000k:11115)****[Ono04]**K. Ono,*The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and -Series*. CBMS Regional Conference Series in Mathematics, 102. American Mathematical Society, Providence, RI, 2004. MR**2020489 (2005c:11053)****[Ono08]**K. Ono,*Unearthing the Visions of a Master: Harmonic Maass Forms and Number Theory*. Harvard-MIT Current Developments in Mathematics 2008, International Press, (preliminary version).**[Shi73]**G. Shimura,*On modular forms of half integral weight*. Ann. of Math. (2) 97 (1973), 440-481. MR**0332663 (48:10989)****[Wal81]**J.-L. Waldspurger,*Sur les coefficients de Fourier des formes modulaires de poids demi-entier*. J. Math. Pures Appl. (9) 60 (1981), no. 4, 375-484. MR**646366 (83h:10061)**

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Additional Information

**John J. Webb**

Affiliation:
Department of Mathematics, University of South Carolina, Columbia, South Caro- lina 29208

Email:
webbjj3@mailbox.sc.edu

DOI:
https://doi.org/10.1090/S0002-9939-09-10188-0

Received by editor(s):
August 28, 2009

Published electronically:
December 8, 2009

Communicated by:
Ken Ono

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.