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Overpartition -rank differences, class number relations, and vector-valued mock Eisenstein series

Volume 189 / 2019

Brandon Williams Acta Arithmetica 189 (2019), 347-365 MSC: Primary 11F37; Secondary 11E41, 11F27. DOI: 10.4064/aa170810-21-10 Published online: 7 June 2019

Abstract

We prove that the generating function of overpartition M2-rank differences is, up to coefficient signs, a component of the vector-valued mock Eisenstein series attached to a certain quadratic form. We use this to compute analogs of the class number relations for M2-rank differences. As applications we split the Kronecker–Hurwitz relation into its “even” and “odd” parts and calculate sums over Hurwitz class numbers of the form \sum_{r \in \mathbb{Z}} H(n - 2r^2).

Authors

  • Brandon WilliamsDepartment of Mathematics
    University of California
    Berkeley, CA 94720, U.S.A.
    and
    Fachbereich Mathematik
    Technische Universität Darmstadt
    64289 Darmstadt, Germany
    e-mail

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