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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: sahlgren@illinois.edu
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: bkim4@seoultech.ac.kr

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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