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Mock theta functions and weakly holomorphic modular forms modulo 2 and 3

Published online by Cambridge University Press:  04 December 2014

SCOTT AHLGREN
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A. e-mail: [email protected]
BYUNGCHAN KIM
Affiliation:
School of Liberal Arts, Seoul National University of Science and Technology, 232 Gongreung-ro, Nowon-gu, Seoul, 139-743, Korea. e-mail: [email protected]

Abstract

We prove that the coefficients of the mock theta functions

\begin{eqnarray*} f(q) = \sum_{n=1}^{\infty} \frac{ q^{n^2}}{(1+q)^2 (1+q^2)^2 \cdots (1+q^n)^2 } \end{eqnarray*}
and
\begin{eqnarray*} \omega(q)=1+\sum_{n=1}^\infty \frac{q^{2n^2+2n}}{(1+q)^2(1+q^3)^2\cdots (1+q^{2n+1})^2} \end{eqnarray*}
possess no linear congruences modulo 3. We prove similar results for the moduli 2 and 3 for a wide class of weakly holomorphic modular forms and discuss applications. This extends work of Radu on the behavior of the ordinary partition function.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2014 

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References

REFERENCES

[1]Ahlgren, S. and Boylan, M.Arithmetic properties of the partition function. Invent. Math. 153 (3) (2003), 487502.Google Scholar
[2]Ahlgren, S. and Boylan, M.Odd coefficients of weakly holomorphic modular forms. Math. Res. Lett. 15 (3) (2008), 409418.Google Scholar
[3]Ahlgren, S. and Ono, K.Congruence properties for the partition function. Proc. Natl. Acad. Sci. USA. 98 (23) (2001), 1288212884.CrossRefGoogle ScholarPubMed
[4]Andersen, N.Classification of congruences for mock theta functions and weakly holomorphic modular forms. Quart. J. Math. 65 (2014), 781805.Google Scholar
[5]Andrews, G. E.On the theorems of Watson and Dragonette for Ramanujan's mock theta functions. Amer. J. Math. 88 (1966), 454490.Google Scholar
[6]Andrews, G. E.Generalised Frobenius partitions. Mem. Amer. Math. Soc. 49 (301) (1984), iv+44.Google Scholar
[7]Andrews, G. E.Partitions, Durfee symbols and the Atkin–Garvan moments of ranks. Invent. Math. 169 (1) (2007), 3773.CrossRefGoogle Scholar
[8]Andrews, G. E.A survey of multipartitions: congruences and identities. In Surveys in Number Theory. Dev. Math. vol. 17. (Springer, New York, 2008), pp. 119.Google Scholar
[9]Andrews, G. E. and Garvan, F. G.Dyson's crank of a partition. Bull. Amer. Math. Soc. (N.S.) 18 (2) (1988), 167171.Google Scholar
[10]Baruah, N. D. and Sarmah, B. K.Congruences for generalised Frobenius partitions with 4 colors. Discrete Math. 311 (17) (2011), 18921902.Google Scholar
[11]Bringmann, K. and Ono, K.The f(q) mock theta function conjecture and partition ranks. Invent. Math. 165 (2) (2006), 243266.Google Scholar
[12]Bruinier, J. H. and Funke, J.On two geometric theta lifts. Duke Math J. 125 (1) (2004), 4590.Google Scholar
[13]Chan, H.-C.Ramanujan's cubic continued fraction and an analog of his “most beautiful identity”. Int. J. Number Theory. 6 (3) (2010), 673680.Google Scholar
[14]Choi, D., Kang, S.-Y. and Lovejoy, J.Partitions weighted by the parity of the crank. J. Combin. Theory Ser. A. 116 (5) (2009), 10341046.Google Scholar
[15]Deligne, P. and Rapoport, M.Les schémas de modules de courbes elliptiques. In Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972). Lecture Notes in Math., Vol. 349. (Springer, Berlin, 1973), pp. 143316.Google Scholar
[16]Diamond, F. and Im, J.Modular forms and modular curves. In Seminar on Fermat's Last Theorem (Toronto, ON, 1993–1994). vol. 17 CMS Conf. Proc.. (Amer. Math. Soc., Providence, RI, 1995), pp, 39133.Google Scholar
[17]Folsom, A., Kent, Z. A. and Ono, K.ℓ-adic properties of the partition function. Adv. Math. 229 (3) (2012), 15861609. Appendix A by Nick Ramsey.Google Scholar
[18]Garvan, F. G.Biranks for partitions into 2 colors. In Ramanujan Rediscovered. vol. 14. Ramanujan Math. Soc. Lect. Notes Ser. (Ramanujan Math. Soc., Mysore, 2010), pp. 87111.Google Scholar
[19]Hammond, P. and Lewis, R.Congruences in ordered pairs of partitions. Int. J. Math. Math. Sci. (45–48) (2004), 25092512.Google Scholar
[20]Hirschhorn, M. D. and Sellers, J. A.Two congruences involving 4-cores. Electron. J. Combin. 3 (2): Research Paper 10, approx. 8 pp. (electronic) (1996). The Foata Festschrift.Google Scholar
[21]Lewis, R.The components of modular forms. J. London Math. Soc. (2). 52 (2) (1995), 245254.Google Scholar
[22]Lovejoy, J.Ramanujan-type congruences for three colored Frobenius partitions. J. Number Theory. 85 (2) (2000), 283290.Google Scholar
[23]Newman, M.Construction and application of a class of modular functions ii. Proc. London Math. Soc. (3). 9 (1959), 373387.Google Scholar
[24]Nicolas, J.-L.Parité des valeurs de la fonction de partition p(n) et anatomie des entiers. In Anatomy of Integers. vol. 46. CRM Proc. Lecture Notes. (Amer. Math. Soc., Providence, RI, 2008), pp. 97113.Google Scholar
[25]Ono, K.Distribution of the partition function modulo m. Ann. of Math. (2). 151 (1) (2000), 293307.CrossRefGoogle Scholar
[26]Ono, K.Unearthing the visions of a master: harmonic Maass forms and number theory. In Current Developments in Mathematics, 2008. (Int. Press, Somerville, MA, 2009), pp. 347454.Google Scholar
[27]Parkin, T. R. and Shanks, D.On the distribution of parity in the partition function. Math. Comp. 21 (1967), 466480.Google Scholar
[28]Paule, P. and Radu, C.-S.The Andrews–Sellers family of partition congruences. Adv. Mathematics 230 (3) (2012), 819838.Google Scholar
[29]Radu, C.-S.Proof of a conjecture by Ahlgren and Ono on the non-existence of certain partition congruences. Trans. Amer. Math. Soc. 365 (9) (2013), 48814894.Google Scholar
[30]Radu, C.-S.A proof of Subbarao's conjecture. J. Reine Angew. Math. 672 (2012), 161175.Google Scholar
[31]Subbarao, M. V.Some remarks on the partition function. Amer. Math. Monthly 73 (1966), 851854.Google Scholar
[32]Treneer, S.Congruences for the coefficients of weakly holomorphic modular forms. Proc. London Math. Soc. (3) 93 (2) (2006), 304324.Google Scholar
[33]Treneer, S.Quadratic twists and the coefficients of weakly holomorphic modular forms. J. Ramanujan Math. Soc. 23 (3) (2008), 283309.Google Scholar
[34]Zagier, D. Ramanujan's mock theta functions and their applications (after Zwegers and Ono–Bringmann). Astérisque. (326). Exp. No. 986, vii–viii, 143–164 (2010), 2009. (Séminaire Bourbaki. Vol. 2007/2008).Google Scholar
[35]Zwegers, S. P.Mock θ-functions and real analytic modular forms. In q-series with Applications to Combinatorics, Number Theory and Physics (Urbana, IL, 2000). vol. 291. Contemp. Math.. (Amer. Math. Soc., Providence, RI, 2001), pp. 269277.Google Scholar