Lacunarity of certain partitiontheoretic generating functions
Authors:
Emily Clader, Yvonne Kemper and Matt Wage
Journal:
Proc. Amer. Math. Soc. 137 (2009), 29592968
MSC (2000):
Primary 11F30, 11P82, 11F11; Secondary 11F20
Published electronically:
May 6, 2009
MathSciNet review:
2506454
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We consider a certain family of infinite products, denoted , which were introduced by Han as a generalization of the NekrasovOkounkov formula. Extending the work of Serre on powers of Dedekind's function, we investigate the integers and for which ``almost all'' of the Fourier coefficients of are zero (forms with this property are referred to as lacunary). We give the complete list of pairs , where is odd, for which is lacunary.
 1.
Tom
M. Apostol, Modular functions and Dirichlet series in number
theory, 2nd ed., Graduate Texts in Mathematics, vol. 41,
SpringerVerlag, New York, 1990. MR 1027834
(90j:11001)
 2.
Pierre
Deligne and JeanPierre
Serre, Formes modulaires de poids 1, Ann. Sci. École
Norm. Sup. (4) 7 (1974), 507–530 (1975) (French). MR 0379379
(52 #284)
 3.
Frank
Garvan, Dongsu
Kim, and Dennis
Stanton, Cranks and 𝑡cores, Invent. Math.
101 (1990), no. 1, 1–17. MR 1055707
(91h:11106), http://dx.doi.org/10.1007/BF01231493
 4.
GN. Han, The NekrasovOkounkov hook length formula: refinement, elementary proof, extension, and applications (preprint).
 5.
Henryk
Iwaniec, Topics in classical automorphic forms, Graduate
Studies in Mathematics, vol. 17, American Mathematical Society,
Providence, RI, 1997. MR 1474964
(98e:11051)
 6.
Neal
Koblitz, Introduction to elliptic curves and modular forms,
2nd ed., Graduate Texts in Mathematics, vol. 97, SpringerVerlag, New
York, 1993. MR
1216136 (94a:11078)
 7.
Ken
Ono, Gordon’s 𝜀conjecture on the lacunarity of
modular forms, C. R. Math. Acad. Sci. Soc. R. Can. 20
(1998), no. 4, 103–107 (English, with English and French
summaries). MR
1662100 (99m:11055)
 8.
Ken
Ono, The web of modularity: arithmetic of the coefficients of
modular forms and 𝑞series, CBMS Regional Conference Series in
Mathematics, vol. 102, Published for the Conference Board of the
Mathematical Sciences, Washington, DC; by the American Mathematical
Society, Providence, RI, 2004. MR 2020489
(2005c:11053)
 9.
JP. Serre, Quelques applications du théorème de densité de Chebatorev, Publ. Math. I.H.E.S. 54 (1981), pp. 123201.
 10.
JeanPierre
Serre, Sur la lacunarité des puissances de 𝜂,
Glasgow Math. J. 27 (1985), 203–221 (French). MR 819840
(87e:11066), http://dx.doi.org/10.1017/S0017089500006194
 1.
 T. Apostol, Modular Functions and Dirichlet Series in Number Theory, Graduate Texts in Mathematics, Second Edition, SpringerVerlag, New York, 1997. MR 1027834 (90j:11001)
 2.
 P. Deligne and J.P. Serre, Formes modulaires de poids 1, Ann. Sci. Éc. Norm. Sup. 7 (1974), pp. 507530. MR 0379379 (52:284)
 3.
 F. Garvan, D. Kim, D. Stanton, Cranks and cores, Invent. Math. 101 (1990), pp. 117. MR 1055707 (91h:11106)
 4.
 GN. Han, The NekrasovOkounkov hook length formula: refinement, elementary proof, extension, and applications (preprint).
 5.
 H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, Vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964 (98e:11051)
 6.
 N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Mathematics, Second Edition, SpringerVerlag, New York, 1993. MR 1216136 (94a:11078)
 7.
 K. Ono, Gordon's conjecture on the lacunarity of modular forms, Comptes Rendus Math. Rep. of Acad. Sci. Canada 20, No. 4 (1998), pp. 103107. MR 1662100 (99m:11055)
 8.
 K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and qseries, CMBS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 2004. MR 2020489 (2005c:11053)
 9.
 JP. Serre, Quelques applications du théorème de densité de Chebatorev, Publ. Math. I.H.E.S. 54 (1981), pp. 123201.
 10.
 JP. Serre, Sur la lacunarité des puissances de , Glasgow Math. J. 27 (1985), pp. 203221. MR 819840 (87e:11066)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
11F30,
11P82,
11F11,
11F20
Retrieve articles in all journals
with MSC (2000):
11F30,
11P82,
11F11,
11F20
Additional Information
Emily Clader
Affiliation:
Department of Mathematics, Columbia University, New York, New York 10027
Yvonne Kemper
Affiliation:
Department of Mathematics, University of California Berkeley, Berkeley, California 94708
Matt Wage
Affiliation:
Appleton East High School, 1411 N. Briarcliff Drive, Appleton, Wisconsin 54915
Address at time of publication:
Princeton University, 0920 Frist Campus Center, Princeton, New Jersey 08544
DOI:
http://dx.doi.org/10.1090/S0002993909098967
PII:
S 00029939(09)098967
Received by editor(s):
July 31, 2008
Received by editor(s) in revised form:
January 19, 2009
Published electronically:
May 6, 2009
Communicated by:
Jim Haglund
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
