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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  and
BYUNGCHAN KIM
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]
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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
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 158 , Issue 1 , January 2015 , pp. 111 - 129
Copyright
Copyright © Cambridge Philosophical Society 2014 

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