The divisor function along sums of two biquadrates

TL;DR

This paper proves improved asymptotic estimates for the sum of the divisor function evaluated along a binary quartic form, utilizing advanced analytic techniques and automorphic forms.

Contribution

It introduces new power-saving asymptotics for divisor sums along binary quartic forms, extending previous results with novel methods.

Findings

Established power-saving asymptotics for divisor sums along binary quartic forms

Applied a recent two-dimensional delta method in this context

Utilized automorphic forms from cubic Dedekind zeta functions

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 $\mathrm{GL}_2$ automorphic forms arising from the factorization of varying cubic Dedekind zeta functions.