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Multidimensional small divisor functions

Andreas Mono

Abstract

This is a short note generalizing the construction from arXiv:1906.07410, arXiv:2009.04955 to multi-indices. We recommend to consider both references first. We obtain polar harmonic Maaß forms of non-positive integral weight if the dimension is even and greater than $2$. We provide explicit examples in dimension $4$, $6$, $8$, and $10$.

Multidimensional small divisor functions

Abstract

This is a short note generalizing the construction from arXiv:1906.07410, arXiv:2009.04955 to multi-indices. We recommend to consider both references first. We obtain polar harmonic Maaß forms of non-positive integral weight if the dimension is even and greater than . We provide explicit examples in dimension , , , and .

Paper Structure

This paper contains 12 sections, 8 theorems, 39 equations.

Table of Contents

  1. Introduction - One-dimensional case
  2. Statement of the result
  3. Multidimensional Case
  4. First step
  5. Second step
  6. Third step
  7. Conclusion
  8. Numerical examples
  9. An interlude on Jacobi polynomials
  10. Explicit examples
  11. Higher even dimensions
  12. Higher odd dimensions

Key Result

Theorem 1.2

Suppose that $\psi = \chi \neq \mathbbm{1}$, and that $P_2\left(\frac{n}{d},d\right) = d.$ Denote the corresponding small divisor function by $\sigma^{\mathrm{sm}}_{1}$, and by $E_2$ the Eisenstein series Define where $\alpha_{\psi}$ is an implicit constant depending only on $\psi$ to ensure a certain growth condition. Then the function $\mathcal{E}^+ + \mathcal{E}^-$ is a polar harmonic Maaß fo

Theorems & Definitions (13)

  • Definition 1.1
  • Theorem 1.2: mertens2019mock
  • Theorem 1.3: mamoro
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • ...and 3 more