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Representations of integers as sums of four polygonal numbers and partial theta functions

Kathrin Bringmann, Min-Joo Jang, Ben Kane, Cheuk Hin Alvin Tse

Abstract

In this paper, we consider representations of integers as sums of at most four distinct $m$-gonal numbers (allowing a fixed number of repeats of each polygonal number occurring in the sum). We show that the number of such representations with non-negative parameters (hence counting the number of points in a regular $m$-gon) is asymptotically the same as $\frac{1}{16}$ times the number of such representations with arbitrary integer parameters (often called generalized polygonal numbers).

Representations of integers as sums of four polygonal numbers and partial theta functions

Abstract

In this paper, we consider representations of integers as sums of at most four distinct -gonal numbers (allowing a fixed number of repeats of each polygonal number occurring in the sum). We show that the number of such representations with non-negative parameters (hence counting the number of points in a regular -gon) is asymptotically the same as times the number of such representations with arbitrary integer parameters (often called generalized polygonal numbers).

Paper Structure

This paper contains 17 sections, 25 theorems, 172 equations.

Table of Contents

  1. Introduction and statement of results
  2. Sums of squares with congruence conditions and polygonal numbers
  3. Basic facts on Farey fractions
  4. Modular Transformations
  5. The theta functions
  6. The false theta functions
  7. Bounding $\mathcal{I}(\mu,k;z)$
  8. Rewriting $\mathcal{I}(\mu,k;z)$
  9. Asymptotics for $\mathcal{I}(\mu,k;z)$
  10. Summing $\mathcal{I}(\mu,k;z)$
  11. Proof of Theorem \ref{['thm:rNrZ']}
  12. Kloosterman's fundamental lemma
  13. Setting up the Circle Method
  14. Bounding $\sum_{\bm{\nu}\in\mathbb N_0^4\setminus\{\bm{0}\}}\bm{I}_{\bm{\nu}}(n)$
  15. Bounding $\bm{I}_{\bm{0}}(n)$
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\bm{\alpha}\in \mathbb N^4$ and $r,M\in\mathbb N$.

Theorems & Definitions (44)

  • Theorem 1.1
  • Remark
  • Corollary 1.2
  • Remark
  • Corollary 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 34 more