Abstract
We consider partial theta series associated with periodic sequences of coefficients, namely, \(\Theta(\tau):= \sum_{n>0} n^\nu f(n) e^{i\pi n^2\tau/M}\), where \(\nu\in\mathbb{Z}_{\ge0}\) and
\(f\colon\mathbb{Z} \to \mathbb{C}\) is an \(M\)-periodic function. Such a function \(\Theta\) is analytic in the half-plane \(\{ \operatorname {Im}\tau>0\}\) and in the asymptotics of \(\Theta(\tau)\) as \(\tau\) tends nontangentially to any \(\alpha\in\mathbb{Q}\) a formal power series appears, which depends on the parity of \(\nu\) and \(f\). We discuss the summability and resurgence properties of these series; namely, we present explicit formulas for their formal Borel transforms and their consequences for the modularity properties of \(\Theta\), or its “quantum modularity” properties in the sense of Zagier’s recent theory. The discrete Fourier transform of \(f\) plays an unexpected role and leads to a number-theoretic analogue of Écalle’s “bridge equations.” The main thesis is: (quantum) modularity \(=\) Stokes phenomenon \(+\) discrete Fourier transform.
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Notes
The convolution product \(*\) is given by the formula \(\hat\psi_1*\hat\psi_2(\xi) = \int_0^\xi \hat\psi_1(\xi_1) \hat\psi_2(\xi-\xi_1)\,d\xi_1\) for \(|\xi|\) small enough and is known to have endless analytic continuation when both factors have (see [6] or [18, Chap. 6])). Of course, \(\mathcal{L}^{\theta'}(\hat\psi_1*\hat\psi_2) = (\mathcal{L}^{\theta'}\hat\psi_1)(\mathcal{L}^{\theta'}\hat\psi_2)\) for a singularity-free direction \(\theta'\), and \(\mathcal{S}^{\theta'}(\tilde\psi_1\tilde\psi_2)=(\mathcal{S}^{\theta'}\tilde\psi_1)(\mathcal{S}^{\theta'}\tilde\psi_2)\) in that case.
By a cusp we mean any orbit of the action of \(\Gamma\) on \(\mathbb{Q}\cup \{i\infty\}\).
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Funding
The second and third authors thank Capital Normal University for hospitality. The second author acknowledges support from NSFC (grant no. 11771303). The fourth author is partially supported by National Key R&D Program of China (grant no. 2020YFA0713300) and by NSFC (grant nos. 11771303, 12171327, 11911530092, and 11871045). This paper is partly a result of the ERC-SyG project, Recursive and Exact New Quantum Theory (ReNewQuantum), which received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 810573).
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2023, Vol. 57, pp. 89–112 https://doi.org/10.4213/faa4031.
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Han, L., Li, Y., Sauzin, D. et al. Resurgence and Partial Theta Series. Funct Anal Its Appl 57, 248–265 (2023). https://doi.org/10.1134/S001626632303005X
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DOI: https://doi.org/10.1134/S001626632303005X