2021/12/30 by Ling Han, Han, Li, Y Li +5 · 1 citation
Mathematics · #Advanced Mathematical Identities #Complex Variables (math.CV) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2112.15223
openalex publication_date 2021/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We consider partial theta series associated with periodic sequences of coefficients, of the form Θ(τ) := ∑n>0 nνf(n) eiπn2τ/M, with ν non-negative integer and an M-periodic function f : ℤ → ℂ. Such a function is analytic in the half-plane \Im(τ)>0\ and as τ tends non-tangentially to any α∈ℚ, a formal power series appears in the asymptotic behaviour of Θ(τ), depending on the parity of ν 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 Θ, 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 motto is: (quantum) modularity = Stokes phenomenon + Discrete Fourier Transform.