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Mathematics 2013, 1(3), 100110; doi:10.3390/math1030100
Article
Effective Congruences for Mock Theta Functions
^{1}
Department of Mathematics, University of Illinois at UrbanaChampaign, 409 W. Green Street, Urbana, IL 61801, USA
^{2}
Department of Mathematics, University of Massachusetts, Lederle Graduate Research Tower, Amherst, MA 01003, USA
^{3}
Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907, USA
^{4}
Department of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, MN 55455, USA
* Author to whom correspondence should be addressed.
Received: 18 July 2013 / Revised: 27 August 2013 / Accepted: 27 August 2013 / Published: 4 September 2013
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Abstract
Let $M\left(q\right)\text{=}\sum {\text{c(n)q}}^{n}$ be one of Ramanujan’s mock theta functions. We establish the existence of infinitely many linear congruences of the form: $c(An\text{}+\text{}B\text{)}\equiv \text{0(mod}{l}^{j}\text{)}$ where A is a multiple of l and an auxiliary prime, p. Moreover, we give an effectively computable upper bound on the smallest such p for which these congruences hold. The effective nature of our results is based on the prior works of Lichtenstein [1] and Treneer [2].Keywords:
mock theta functions; congruences; harmonic weak Maass forms
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