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Annals of Combinatorics

, Volume 23, Issue 2, pp 417–422 | Cite as

On the Zeros of a Class of Modular Functions

  • Naomi SweetingEmail author
  • Katharine Woo
Article
  • 12 Downloads

Abstract

We generalize a number of works on the zeros of certain level 1 modular forms to a class of weakly holomorphic modular functions whose q-expansions satisfy the following:
$$\begin{aligned} f_k(A; \tau )\, {:}{=}\,q^{-k}(1+a(1)q+a(2)q^2+\cdots ) + O(q), \end{aligned}$$
where a(n) are numbers satisfying a certain analytic condition. We show that the zeros of such \(f_k(\tau )\) in the fundamental domain of \(\text {SL}_2(\mathbb {Z})\) lie on \(|\tau |=1\) and are transcendental. We recover as a special case earlier work of Witten on extremal “partition” functions \(Z_k(\tau )\). These functions were originally conceived as possible generalizations of constructions in three-dimensional quantum gravity.

Keywords

Number theory Zeros Modular functions 

Mathematics Subject Classification

11F03 

Notes

Acknowledgements

The authors would like to thank Professors Ken Ono and Larry Rolen for their support, guidance, and suggestions. They also thank Emory University, the Asa Griggs Candler Fund, and NSF grant DMS-1557960.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsStanford UniversityStanfordUSA

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