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Quantum modularity of partial theta series with periodic coefficients

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arxiv 2012.02457 v1 pith:CPXT7UEV submitted 2020-12-04 math.NT math.GTmath.QA

Ankush Goswami , Robert Osburn This is my paper · ORCID
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keywords quantumseriesmodularitycoefficientspartialperiodicthetaapplication
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abstract

We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich-Zagier series $\mathscr{F}_t(q)$ which matches (at a root of unity) the colored Jones polynomial for the family of torus knots $T(3,2^t)$, $t \geq 2$, is a weight $3/2$ quantum modular form. This generalizes Zagier's result on the quantum modularity for the "strange" series $F(q)$.

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Cited by 1 Pith paper

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    Vector-valued modular resurgent series are defined, and Eichler integrals of unary theta series are proven to be non-trivial examples whose Stokes constants are rotated by the modular S-matrix and whose median resumma...

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