Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Resurgence and Partial Theta Series (2112.15223v3)

Published 30 Dec 2021 in math.CV

Abstract: We consider partial theta series associated with periodic sequences of coefficients, of the form $\Theta(\tau) := \sum_{n>0} n\nu f(n) e{i\pi n2\tau/M}$, with $\nu$ non-negative integer and an $M$-periodic function $f : \mathbb{Z} \rightarrow \mathbb{C}$. Such a function is analytic in the half-plane ${Im(\tau)>0}$ and as $\tau$ tends non-tangentially to any $\alpha\in\mathbb{Q}$, a formal power series appears in the asymptotic behaviour of $\Theta(\tau)$, depending on the parity of $\nu$ and $f$. We discuss the summability and resurgence properties of these series by means of explicit formulas for their formal Borel transforms, and the 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 \'Ecalle'sBridge Equations''. The motto is: (quantum) modularity = Stokes phenomenon + Discrete Fourier Transform.

Summary

We haven't generated a summary for this paper yet.