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Sums of class numbers and mixed mock modular forms

Published online by Cambridge University Press:  04 June 2018

KATHRIN BRINGMANN  and
BEN KANE
KATHRIN BRINGMANN
Affiliation:
Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany, e-mail: kbringma@math.uni-koeln.de
BEN KANE
Affiliation:
Department of Mathematics, University of Hong Kong, Pokfulam, Hong Kong, e-mail: bkane@maths.hku.hk
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Abstract

In this paper, we consider sums of class numbers of the type ∑m ≡ a (mod p) H (4n − m2), where p is an odd prime, n ∈ ℕ, and a ∈ ℤ. By showing that these are coefficients of mixed mock modular forms, we obtain explicit formulas. Using these formulas for p = 5 and 7, we then prove a conjecture of Brown et al. in the case that n = ℓ is prime.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 2 , September 2019 , pp. 321 - 333
Copyright
Copyright © Cambridge Philosophical Society 2018 

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Footnotes

†

The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp Foundation.

‡

This research was completed while the second author was a postdoc at the University of Cologne.

References

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