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The divisor function along sums of two biquadrates

Wing Hong Leung, Mayank Pandey

TL;DR

This work establishes power-saving asymptotics for the divisor-sum along the binary quartic form $m^4+n^4$, improving upon previous results by exposing a secondary main term. The authors deploy a novel two-dimensional delta method (a perturbation of the Li–Rydin-Myerson–Vishe framework) in concert with Poisson summation over a quartic number field and leverage factorization of cubic Dedekind zeta functions into GL$_2$ automorphic components to gain cancellation across frequencies. The analysis splits into a balanced case (with $X=X_1X_2$ and near-equal factors) and an unbalanced regime, handling nonzero, partial-zero, and full-zero frequency contributions; local densities and singular series are extracted to form the main term, which cancels the dominant contributions from other parts of the expansion, yielding a power-saving error term. The results not only sharpen the asymptotics for the specific form $m^4+n^4$ but also illustrate a roadmap for extending to general binary quartic forms and to GL$_2$ automorphic coefficients beyond the divisor function. The methods have broader implications for understanding equidistribution and density phenomena in polynomial sequences via automorphic tools.

Abstract

We establish power saving asymptotics for the sum of the divisor function along a binary quartic form, improving on work of Daniel. The proof involves an application of a recent two dimensional delta method due to Li, Rydin-Myerson, and Vishe and an exploitation of $\GL_2 $ automorphic forms arising from the factorization of varying cubic Dedekind zeta functions.

The divisor function along sums of two biquadrates

TL;DR

This work establishes power-saving asymptotics for the divisor-sum along the binary quartic form , improving upon previous results by exposing a secondary main term. The authors deploy a novel two-dimensional delta method (a perturbation of the Li–Rydin-Myerson–Vishe framework) in concert with Poisson summation over a quartic number field and leverage factorization of cubic Dedekind zeta functions into GL automorphic components to gain cancellation across frequencies. The analysis splits into a balanced case (with and near-equal factors) and an unbalanced regime, handling nonzero, partial-zero, and full-zero frequency contributions; local densities and singular series are extracted to form the main term, which cancels the dominant contributions from other parts of the expansion, yielding a power-saving error term. The results not only sharpen the asymptotics for the specific form but also illustrate a roadmap for extending to general binary quartic forms and to GL automorphic coefficients beyond the divisor function. The methods have broader implications for understanding equidistribution and density phenomena in polynomial sequences via automorphic tools.

Abstract

We establish power saving asymptotics for the sum of the divisor function along a binary quartic form, improving on work of Daniel. The proof involves an application of a recent two dimensional delta method due to Li, Rydin-Myerson, and Vishe and an exploitation of automorphic forms arising from the factorization of varying cubic Dedekind zeta functions.
Paper Structure (33 sections, 40 theorems, 418 equations)

This paper contains 33 sections, 40 theorems, 418 equations.

Table of Contents

  1. Introduction and statement of results
  2. Sketch of proof and discussion of argument
  3. Notation
  4. Preliminary lemmas
  5. Polynomial root counts in residue classes
  6. Factorization in a rank $2$ submodule of $\mathcal{O}_K$
  7. Poisson summation over $K$
  8. Factorization and coefficients of cubic Dedekind zeta functions
  9. The two dimensional $\delta$-method
  10. Setup for proof of Theorem \ref{['theorem:cq5o3jikeq']} in the balanced case
  11. Proof of Theorem \ref{['theorem:cq5o3jikeq']}
  12. Structure of paper
  13. Proof of Proposition \ref{['proposition:crb36eebaq']}
  14. Estimating $S_1$
  15. Bounding $S_2$
  16. ...and 18 more sections

Key Result

Theorem 1

There exists $\delta > 0$ such that for $N\geqslant 1$, we have that where and $c_{-1}, c_0$ are such that for $\mathop{\mathrm{Re}}\nolimits (s) > 1$, where $\rho(q) = \# { \left\lbrace x_1, x_2\in \mathbb{Z}/q \mathbb{Z} :q | x_1^4 + x_2^4\right\rbrace}$.

Theorems & Definitions (70)

  • Theorem 1
  • Theorem 2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • Lemma 3.5
  • ...and 60 more